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@article{abid2003grosspitaevskii,
 author = {M. Abid and C. Huepe and S. Metens and C. Nore and C. T. Pham and L. S. Tuckerman and M. E. Brachet},
 doi = {10.1016/j.fluiddyn.2003.09.001},
 journal = {Fluid Dyn. Research},
 pages = {509--544},
 title = {{G}ross-{P}itaevskii dynamics of {B}ose-{E}instein condensates and superfluid turbulence},
 volume = {33},
 year = {2003}
}

@book{ablowitz1992solitions,
 address = {Cambridge},
 author = {M. J. Ablowitz and P. A. Clarkson},
 publisher = {Cambridge Univ. Press},
 title = {{S}olitions, {N}onlinear {E}volution {E}quations and {I}nverse {S}cattering},
 year = {1992}
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@book{abraham1978foundations,
 address = {Reading, Mass.},
 author = {R. Abraham and J. E. Marsden},
 publisher = {Benjamin-Cummings},
 title = {{F}oundations of {M}echanics},
 year = {1978}
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@book{abraham1992dynamics,
 address = {Reading, MA},
 author = {R. H. Abraham and C. D. Shaw},
 publisher = {Addison-Wesley},
 title = {{D}ynamics - {T}he {G}eometry of {B}ehavior},
 year = {1992}
}

@article{abshagen2005mode,
 author = {J. Abshagen and J. M. Lopez and F. Marques and G. Pfister},
 journal = {J. Fluid Mech.},
 pages = {269--300},
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 year = {2005}
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@article{abshagen2005symmetry,
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@article{acarlar1987study,
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 year = {1987}
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@book{adams2003sobolev,
 address = {New York},
 author = {Adams, R. A. and Fournier, J. J. F.},
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 title = {{S}obolev spaces},
 year = {2003}
}

@article{afraimovich1987on,
 author = {V. S. Afraimovich and B. B. Bykov and L. P. Shilnikov},
 journal = {Dokl. Akad. Nauk SSSR},
 keywords = {Lorenz, attractor},
 note = {In Russian},
 pages = {336--339},
 title = {{O}n the appearence and the structure of the {L}orenz attractor},
 volume = {234},
 year = {1987}
}

@inproceedings{ahuja2006templatebased,
 address = {Piscataway, NJ},
 author = {S. Ahuja and I. G. Kevrekidis and C. W. Rowleym},
 booktitle = {American Control Conference, June 2006},
 doi = {10.1109/ACC.2006.1657278},
 pages = {3606--3611},
 publisher = {IEEE},
 title = {{T}emplate-based stabilization of relative equilibria},
 year = {2006}
}

@article{ahuja2007templatebased,
 author = {S. Ahuja and I. G. Kevrekidis and C. W. Rowley},
 doi = {10.1007/s00332-005-0801-7},
 journal = {J. Nonlin. Sci.},
 pages = {109--143},
 title = {{T}emplate-based stabilization of relative equilibria in systems with continuous symmetry},
 volume = {17},
 year = {2007}
}

@misc{aksamentov2020covid19,
 author = {I. Aksamentov and N. Noll and R. Neher},
 title = {C{O}VI{D}-19 {S}cenarios},
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 year = {2020}
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 author = {E. L. Allgower and K. Georg},
 journal = {Lect. Appl. Math.},
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 title = {{N}umerically stable homotopy methods without an extra dimension},
 url = {www.math.colostate.edu/emeriti/georg/homExtra.pdf},
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 year = {1990}
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@book{alligood1996chaos,
 address = {New York},
 author = {K. T. Alligood and T. D. Sauer and J. A. Yorke},
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 publisher = {Springer-Verlag},
 title = {{C}haos: {A}n {I}ntroduction to {D}ynamical {S}ystems},
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@inbook{alligood1997chapter,
 address = {Berlin, Heidelberg},
 author = {K. T. Alligood and T. D. Sauer and J. A. Yorke},
 booktitle = {Chaos: An Introduction to Dynamical Systems},
 doi = {10.1007/978-3-642-59281-2\_10},
 publisher = {Springer Berlin Heidelberg},
 title = {{C}hapter {O}ne: {O}ne-dimensional {M}aps},
 url = {https://doi.org/10.1007/978-3-642-59281-2\_10},
 year = {1997}
}

@inbook{alligood1997stable,
 address = {Berlin, Heidelberg},
 author = {K. T. Alligood and T. D. Sauer and J. A. Yorke},
 booktitle = {Chaos: An Introduction to Dynamical Systems},
 doi = {10.1007/978-3-642-59281-2\_10},
 pages = {399--445},
 publisher = {Springer Berlin Heidelberg},
 title = {{S}table {M}anifolds and {C}rises},
 url = {https://doi.org/10.1007/978-3-642-59281-2\_10},
 year = {1997}
}

@article{altmann2009poincare,
 abstract = {Simulating observational and experimental
situations, we consider a leak in the phase space of a chaotic
dynamical system. We obtain an expression for the escape rate
of the survival probability applying the theory of transient
chaos. This expression improves previous estimates based on the
properties of the closed system and explains dependencies on
the position and size of the leak and on the initial ensemble.
With a subtle choice of the initial ensemble, we obtain an
equivalence to the classical problem of Poincare recurrences in
closed systems, which is treated in the same framework.
Finally, we show how our results apply to weakly chaotic
systems and justify a split of the invariant saddle in
hyperbolic and non-hyperbolic components, related,
respectively, to the intermediate exponential and asymptotic
power-law decays of the survival probability.},
 archiveprefix = {arXiv},
 author = {Altmann, E. G. and T{\'e}l, T.},
 eprint = {0808.3785},
 journal = {Phys. Rev. E},
 numpages = {12},
 pages = {016204},
 title = {{P}oincar{\'e} recurrences and transient chaos in systems with leaks},
 volume = {79},
 year = {2009}
}

@article{ambrose2009global,
 archiveprefix = {arXiv},
 author = {Ambrose, D. M. and Wilkening, J.},
 doi = {10.2140/camcos.2009.4.177},
 eprint = {0811.4205},
 journal = {Commun. App. Math. and Comp. Sci.},
 pages = {177--215},
 title = {{G}lobal paths of time-periodic solutions of the {B}enjamin-{O}no equation connecting pairs of traveling waves},
 volume = {4},
 year = {2009}
}

@article{ambrose2010computation,
 archiveprefix = {arXiv},
 author = {Ambrose, D. M. and Wilkening, J.},
 doi = {10.1007/s00332-009-9058-x},
 eprint = {0804.3623},
 journal = {J. Nonlinear Science},
 pages = {277--308},
 title = {{C}omputation of time-periodic solutions of the {B}enjamin-{O}no equation},
 volume = {20},
 year = {2010}
}

@article{amigo2010permutation,
 author = {J. M. Amig{\'o} and S. Zambrano and M. A. F. Sanju{\'a}n},
 doi = {10.1209/0295-5075/90/10007},
 journal = {Europhys. Lett.},
 number = {1},
 pages = {10007},
 title = {{P}ermutation complexity of spatiotemporal dynamics},
 volume = {90},
 year = {2010}
}

@article{andersen2015doubleslit,
 author = {A. Andersen and J. Madsen and C. Reichelt and S. Rosenlund Ahl and B. Lautrup and C. Ellegaard and M. T. Levinsen and T. Bohr},
 doi = {10.1103/PhysRevE.92.013006},
 issue = {1},
 journal = {Phys. Rev. E},
 numpages = {14},
 pages = {013006},
 publisher = {American Physical Society},
 title = {{D}ouble-slit experiment with single wave-driven particles and its relation to quantum mechanics},
 volume = {92},
 year = {2015}
}

@article{anderson1972more,
 author = {Anderson, P. W.},
 doi = {10.1126/science.177.4047.393},
 issn = {0036-8075},
 journal = {Science},
 number = {4047},
 pages = {393--396},
 publisher = {American Association for the Advancement of Science},
 title = {{M}ore {I}s {D}ifferent},
 volume = {177},
 year = {1972}
}

@book{anderson1999lapack,
 address = {Philadelphia},
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 edition = {Third},
 isbn = {0-89871-447-8 (paperback)},
 publisher = {Society for Industrial and Applied Mathematics},
 title = {L{A}PA{C}K {U}sers\textquotesingle           {G}uide},
 year = {1999}
}

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 author = {N. Andersson and G. L. Comer},
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 journal = {Living Rev. Relat.},
 title = {{R}elativistic fluid dynamics: {P}hysics for many different scales},
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 year = {2007}
}

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 title = {{T}he periodic orbits of an area preserving twist-map},
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 year = {1988}
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@book{anosov1988dynamical,
 address = {New York},
 author = {D. V. Anosov and V. I. Arnol\textquotesingle          d},
 publisher = {Springer},
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}

@article{antoulas2001lyapunov,
 author = {Antoulas, A. C. and Sorensen, D. C.},
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 year = {2002}
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 author = {G. B. Arfken and H. J. Weber},
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 title = {{M}athematical {M}ethods for {P}hysicists: {A} Comprehensive {G}uide},
 year = {2005}
}

@article{armbruster1988heteroclinic,
 abstract = {Consider the normal form on the center manifold for
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 year = {1988}
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@article{arnold1991kolmogorovtextquotesingle,
 abstract = {A review of Kolmogorov\textquotesingle          s efforts relating
the Navier-Stokes equation to the theory of dynamical system.
Several interesting questions regarding the connection are exposed.},
 author = {V. I. Arnold},
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 title = {{K}olmogorov\textquotesingle          s hydrodynamic attractors},
 volume = {434},
 year = {1991}
}

@book{arnoltextquotesingled1992ordinary,
 address = {New York},
 author = {V. I. Arnol\textquotesingle          d},
 publisher = {Springer},
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 year = {1992}
}

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@article{artuso1990recyclingb,
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 year = {1990}
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@article{artuso2003cycle,
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 title = {{C}ycle expansions for intermittent maps},
 volume = {150},
 year = {2003}
}

@incollection{artuso2013chapter,
 address = {Copenhagen},
 author = {R. Artuso and R. Hugh and P. Cvitanovi{\' c}},
 booktitle = {Chaos: Classical and Quantum},
 editor = {P. Cvitanovi{\' c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
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 publisher = {Niels Bohr Inst.},
 title = {{C}hapter "{W}hy does it work?"},
 year = {2013}
}

@article{ascher1995implicitexplicit,
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 volume = {32},
 year = {1995}
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@article{ascher1997implicitexplicit,
 author = {U. M. Ascher and S. J. Ruuth and R. J. Spiteri},
 journal = {Appl. Num. Math.},
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 title = {{I}mplicit-explicit {R}unge-{K}utta methods for time-dependent partial differential equations},
 volume = {25},
 year = {1997}
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@article{atay2010symbolic,
 author = {F. M. Atay and S. Jalan and J. Jost},
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 pages = {130--135},
 title = {{S}ymbolic dynamics and synchronization of coupled map networks with multiple delays},
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@article{aubry1988dynamics,
 author = {N. Aubry and P. Holmes and J. L. Lumley and E. Stone},
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 year = {1988}
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@article{aubry1995antiintegrability,
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 volume = {86},
 year = {1995}
}

@article{auerbach1987exploring,
 abstract = {Use near recurrence to extract periodic orbits
and their stability eigenvalues.},
 author = {D. Auerbach and P. Cvitanovi{\' c} and J.-P. Eckmann and G. Gunaratne and I. Procaccia},
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 journal = {Phys. Rev. Lett.},
 pages = {23},
 title = {{E}xploring chaotic motion through periodic orbits},
 volume = {58},
 year = {1987}
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@article{avila2008global,
 author = {Avila, M. and Grimes, M. and Lopez, J. M. and Marques, F.},
 doi = {10.1063/1.2996326},
 journal = {Phys. Fluids},
 number = {104104},
 title = {{G}lobal endwall effects on centrifugally stable flows},
 volume = {20},
 year = {2008}
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@article{avila2010on,
 author = {Avila, M. and Willis, A. P. and Hof, B.},
 doi = {10.1017/S0022112009993296},
 journal = {J. Fluid Mech.},
 pages = {127--136},
 title = {{O}n the transient nature of localized pipe flow turbulence},
 volume = {646},
 year = {2010}
}

@article{avila2011onset,
 author = {Avila, K. and Moxey, D. and de Lozar, A. and Avila, M. and Barkley, D. and Hof, B.},
 doi = {10.1126/science.1203223},
 journal = {Science},
 pages = {192--196},
 title = {{T}he onset of turbulence in pipe flow},
 volume = {333},
 year = {2011}
}

@article{avila2012stability,
 author = {Avila, M.},
 doi = {10.1103/PhysRevLett.108.124501},
 journal = {Phys. Rev. Lett.},
 pages = {124501},
 title = {{S}tability and angular-momentum transport of fluid flows between corotating cylinders},
 volume = {108},
 year = {2012}
}

@article{avila2013nature,
 author = {Avila, M. and Hof, B.},
 doi = {10.1103/PhysRevE.87.063012},
 journal = {Phys. Rev. E},
 pages = {063012},
 publisher = {APS},
 title = {{N}ature of laminar-turbulence intermittency in shear flows},
 volume = {87},
 year = {2013}
}

@article{avila2013streamwiselocalized,
 author = {Avila, M. and Mellibovsky, F. and Roland, N. and Hof, B.},
 doi = {10.1103/PhysRevLett.110.224502},
 journal = {Phys. Rev. Lett.},
 pages = {224502},
 title = {{S}treamwise-localized solutions at the onset of turbulence in pipe flow},
 volume = {110},
 year = {2013}
}

@article{axenides2009nambu,
 archiveprefix = {arXiv},
 author = {Axenides, M. and Floratos, E.G. and Nicolis, S.},
 doi = {10.1088/1751-8113/42/27/275201},
 eprint = {0901.2638},
 journal = {J. Phys. A},
 pages = {275201},
 title = {{N}ambu quantum mechanics on discrete 3-tori},
 volume = {42},
 year = {2009}
}

@article{axenides2009nambulie,
 archiveprefix = {arXiv},
 author = {Axenides, M. and Floratos, E.},
 doi = {10.1088/1126-6708/2009/02/039},
 eprint = {0809.3493},
 journal = {JHEP},
 pages = {39},
 title = {{N}ambu-{L}ie 3-algebras on fuzzy 3\textbackslash          mbox-manifolds},
 volume = {902},
 year = {2009}
}

@article{axenides2010strange,
 archiveprefix = {arXiv},
 author = {Axenides, M. and Floratos, E.},
 doi = {10.1007/JHEP04(2010)036},
 eprint = {0910.3881},
 journal = {JHEP},
 pages = {036},
 title = {{S}trange attractors in dissipative {N}ambu mechanics: {C}lassical and quantum aspects},
 volume = {1004},
 year = {2010}
}

@incollection{axenides2011non,
 addendum = {http://arXiv.org/abs/1109.0470},
 author = {Axenides, M.},
 booktitle = {Chaos Theory: Modeling, Simulation and Applications},
 editor = {Skiadas, C. H. and Dimotikalis, I. and Skiadas, C.},
 isbn = {9789814460477},
 publisher = {World Scientific},
 title = {{N}on {H}amiltonian chaos from {N}ambu dynamics of surfaces},
 year = {2011}
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@article{babaee2016minimization,
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 author = {Bakewell, P. and Lumley, J. L.},
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@article{baladi1996sharp,
 author = {Baladi, V. and Ruelle, D.},
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@article{baladi1997sharp,
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 title = {{S}harp determinants and kneading operators for holomorphic maps},
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@inproceedings{balay1997efficient,
 author = {S. Balay and W. D. Gropp and L. C. McInnes and B. F. Smith},
 booktitle = {Modern Software Tools in Scientific Computing},
 editor = {E. Arge and A. M. Bruaset and H. P. Langtangen},
 pages = {163--202},
 publisher = {Birkh{\"a}user Press},
 title = {{E}fficient management of parallelism in object oriented numerical software libraries},
 year = {1997}
}

@misc{balay2001petsc,
 author = {S. Balay and K. Buschelman and W. D. Gropp and D. Kaushik and M. G. Knepley and L. C. McInnes and B. F. Smith and H. Zhang},
 note = {www.mcs.anl.gov/petsc},
 title = {P{E}TSc web page},
 year = {2001}
}

@techreport{balay2004petsc,
 author = {S. Balay and K. Buschelman and V. Eijkhout and W. D. Gropp and D. Kaushik and M. G. Knepley and L. C. McInnes and B. F. Smith and H. Zhang},
 institution = {Argonne National Laboratory},
 number = {ANL-95/11 - Revision 2.1.5},
 title = {P{E}TSc {U}sers {M}anual},
 year = {2004}
}

@article{bandelowrecke1998frequency,
 author = {U. Bandelow, L. Recke and B. Sandstede},
 journal = {Opt. Commun.},
 pages = {212--218},
 title = {{F}requency regions for forced locking of self-pulsating multi-section {D}FB lasers},
 volume = {147},
 year = {1998}
}

@article{barabasi1999emergence,
 author = {A.-L. Barab{\'a}si and R. Albert},
 journal = {Science},
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 pages = {509--512},
 title = {{E}mergence of scaling in random networks},
 volume = {286},
 year = {1999}
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@article{baranger1988calculation,
 author = {M. Baranger and K. T. R. Davies and J. H. Mahoney},
 doi = {10.1016/S0003-4916(88)80018-6},
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@article{barashenkov2011timeperiodic,
 author = {Barashenkov, I. V. and Zemlyanaya, E. V. and van Heerden, T. C.},
 doi = {10.1103/PhysRevE.83.056609},
 journal = {Phys. Rev. E},
 pages = {056609},
 title = {{T}ime-periodic solitons in a damped-driven nonlinear {S}chr{\"o}dinger equation},
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 author = {E. Ben-Naim and P. L. Krapivsky},
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@incollection{beyn2014stability,
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 editors = {Dieci, L. and Guglielmi, N.},
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 publisher = {Springer},
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@article{birkhoff1935nouvelles,
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@book{biro1994chaos,
 address = {Singapore},
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@book{bleistein1986asymptotic,
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@inproceedings{bodenschatz2011prandtl,
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 booktitle = {Highlights in the History of Turbulence},
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 title = {{P}randtl and the {G}{\"o}ttingen {S}chool},
 year = {2011}
}

@book{bohr1998dynamical,
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 year = {1998}
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@article{bolhuis2002transition,
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 journal = {Annu. Rev. Phys. Chem},
 number = {1},
 pages = {291-318},
 title = {T{R}AN{S}IT{I}ON {P}AT{H} S{A}MP{L}IN{G}: {T}hrowing {R}opes {O}ver {R}ough {M}ountain {P}asses, in the {D}ark},
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 year = {2002}
}

@article{bollt2007attractor,
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 year = {2007}
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@article{borreroecheverry2010transient,
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@article{bottin1998discontinuous,
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 doi = {10.1140/epje/i2014-14034-1},
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 title = {{D}iscontinuous transition to spatiotemporal intermittency in plane {C}ouette flow},
 volume = {43},
 year = {1998}
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@incollection{bottin1998discontinuousb,
 author = {Bottin, S. and Daviaud, F. and Manneville, P. and Dauchot, O.},
 booktitle = {Advances in Turbulence VII},
 doi = {10.1007/978-94-011-5118-4\_28},
 editor = {Frisch, U.},
 isbn = {978-94-010-6151-3},
 pages = {117--120},
 publisher = {Springer},
 title = {{D}iscontinuous {T}ransition to {S}patio-{T}emporal {I}ntermittency in the {P}lane {C}ouette {F}low},
 year = {1998}
}

@article{bottin1998experimental,
 author = {Bottin, S. and Dauchot, O. and Daviaud, F. and Manneville, P.},
 doi = {10.1063/1.869773},
 journal = {Phys. Fluids},
 pages = {2597--2607},
 title = {{E}xperimental evidence of streamwise vortices as finite amplitude solutions in transitional plane {C}ouette flow},
 volume = {10},
 year = {1998}
}

@article{bottin1998statistical,
 author = {Bottin, S. and Chat{\'e}, H.},
 doi = {10.1007/s100510050536},
 journal = {Eur. Phys. J. B},
 pages = {143--155},
 title = {{S}tatistical analysis of the transition to turbulence in plane {C}ouette flow},
 volume = {6},
 year = {1998}
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@unpublished{bouchbinder2003nonlinear,
 author = {E. Bouchbinder},
 note = {Unpublished notes},
 title = {{T}he nonlinear {S}chr{\"o}dinger equation},
 url = {http://ChaosBook.org/library/Bouch03.pdf},
 year = {2003}
}

@article{bouquet1995hamiltonian,
 abstract = {The stationary KSe can be transformed into a time-depent
1-d Hamiltonian system. There exist a large class of initial
conditions under which the system will explode in a finite time like
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 author = {S. Bouquet},
 journal = {J. Math. Phys.},
 pages = {1242},
 title = {{H}amiltonian structure and integrability of the stationary {K}uramoto-{S}ivashinsky equation},
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 year = {1995}
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@book{bowen1975equilibrium,
 address = {Berlin},
 author = {R. Bowen},
 publisher = {Springer},
 title = {{E}quilibrium {S}tates and the {E}rgodic {T}heory of {A}nosov {D}iffeomorphisms},
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@book{boyce2006elementary,
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 author = {W. E. Boyce and R. C. DiPrima},
 publisher = {Wiley-Interscience},
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 year = {2006}
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@article{brand2014doubly,
 archiveprefix = {arXiv},
 author = {E. Brand and J. F. Gibson},
 doi = {10.1017/jfm.2014.285},
 eprint = {1404.2887},
 journal = {J. Fluid Mech.},
 pages = {R3},
 title = {{A} doubly localized equilibrium solution of plane {C}ouette flow},
 volume = {750},
 year = {2014}
}

@incollection{brandenburg2003computational,
 address = {London},
 archiveprefix = {arXiv},
 author = {A. Brandenburg},
 booktitle = {Advances in Nonlinear Dynamos},
 editor = {A. Ferriz-Mas and M. N{\'u}{\~n}ez},
 eprint = {astro-ph0109497},
 publisher = {Taylor \& Francis},
 title = {{C}omputational aspects of astrophysical {M}HD and turbulence},
 year = {2003}
}

@misc{brandenburg2020quadratic,
 archiveprefix = {arXiv},
 author = {A. Brandenburg},
 eprint = {2002.03638},
 primaryclass = {q-bio.PE},
 title = {{Q}uadratic growth during the 2019 novel coronavirus epidemic},
 year = {2020}
}

@article{brandt2014liftup,
 archiveprefix = {arXiv},
 author = {Brandt, L.},
 eprint = {1403.4657},
 journal = {Eur. J. Mech.-B/Fluids},
 pages = {80--96},
 publisher = {Elsevier},
 title = {{T}he lift-up effect: the linear mechanism behind transition and turbulence in shear flows},
 volume = {47},
 year = {2014}
}

@article{bridges2008degenerate,
 abstract = {A fundamental class of solutions of symmetric Hamiltonian systems
is relative equilibria. In this paper the nonlinear problem near
a degenerate relative equilibrium is considered. The degeneracy creates
a saddle-center and attendant homoclinic bifurcation in the reduced
system transverse to the group orbit. The surprising result is that
the curvature of the pullback of the momentum map to the Lie algebra
determines the normal form for the homoclinic bifurcation. There
is also an induced directional geometric phase in the homoclinic
bifurcation. The backbone of the analysis is the use of singularity
theory for smooth mappings between manifolds applied to the pullback
of the momentum map. The theory is constructive and generalities
are given for symmetric Hamiltonian systems on a vector space of
dimension (2n+2) with an n-dimensional abelian symmetry group. Examples
for n=1,2,3 are presented to illustrate application of the theory.},
 author = {T. J. Bridges},
 journal = {J. Diff. Eqn.},
 pages = {1629--1674},
 title = {{D}egenerate relative equilibria, curvature of the momentum map,and homoclinic bifurcation},
 volume = {244},
 year = {2008}
}

@misc{bridgesnoyearnotitle,
 author = {T. J. Bridges},
 note = {private communication}
}

@article{briggs1991feigenvalues,
 author = {K. M. Briggs and G. R. W. Quispel and C. J. Thompson},
 journal = {J. Phys. A},
 keywords = {Feigenbaum constant, Mandelbrot set},
 pages = {3363--3368},
 title = {{F}eigenvalues for {M}andelsets},
 volume = {24},
 year = {1991}
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@article{briggs1991precise,
 author = {Briggs, K.},
 journal = {Math. Computation},
 pages = {435--439},
 title = {{A} precise calculation of the {F}eigenbaum constants},
 volume = {57},
 year = {1991}
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@article{briggs1998analytic,
 author = {Briggs, K. M. and Dixon, T. W. and Szekeres, G.},
 journal = {Int. J. Bif. Chaos},
 pages = {347--357},
 title = {{A}nalytic solutions of the {C}vitanovic-{F}eigenbaum and {F}eigenbaum-{K}adanoff-{S}henker equations},
 volume = {8},
 year = {1998}
}

@article{brockmann2006scaling,
 author = {D. Brockmann and L. Hufnagel and T. Geisel},
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 journal = {Nature},
 number = {7075},
 pages = {462--465},
 title = {{T}he scaling laws of human travel},
 volume = {439},
 year = {2006}
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@article{brodin1994nonlinear,
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 title = {{N}onlinear surface waves in a plasma with a diffuse boundary},
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 year = {1994}
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@article{broer1997algorithms,
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 journal = {Z. Angew. Math. Phys.},
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 year = {1997}
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@article{bronski2006uncertainty,
 archiveprefix = {arXiv},
 author = {J. C. Bronski and T. N. Gambill},
 eprint = {math/0508481},
 journal = {Nonlinearity},
 pages = {2023--2039},
 title = {{U}ncertainty estimates and \textdollar          {L}\_2\textdollar           bounds for the {K}uramoto-{S}ivashinsky equation},
 volume = {19},
 year = {2006}
}

@inproceedings{brown1996modulated,
 address = {Providence, RI},
 author = {Brown, H. S. and Kevrekidis, I. G.},
 booktitle = {Pattern Formation: Symmetry Methods and Applications},
 editor = {D. Benest and C. Froeschl{\'e}},
 pages = {45--66},
 publisher = {AMS},
 series = {Fields Inst. Commun.},
 title = {{M}odulated traveling waves for the {K}uramoto-{S}ivashinsky equation},
 volume = {5},
 year = {1996}
}

@article{brusch2000modulated,
 abstract = {MAWs of various periods P occur in phase chaotic states.
The pair of MAWs cease to exist via a saddle-node bifurcation. For
periods beyond this bifurcation, incoherent near-MAW evolve toward
defects.},
 author = {L. Brusch and M. G. Zimmermann and M. van Hecke and M. B{\"a}r and A. Torcini},
 journal = {Phys. Rev. Lett.},
 pages = {86},
 title = {{M}odulated amplitude waves and the transition from phase to defect chaos},
 volume = {85},
 year = {2000}
}

@article{brusch2001modulated,
 author = {L. Brusch and A. Torcini and M. van Hecke and M. G. Zimmermann and M. B{\"a}r},
 journal = {Physica D},
 pages = {127--148},
 title = {{M}odulated amplitude waves and defect formation in the one-dimensional complex {G}inzburg-{L}andau equation},
 volume = {160},
 year = {2001}
}

@article{brusch2003nonlinear,
 author = {L. Brusch and A. Torcini and M. B{\"a}r},
 journal = {Physica D},
 pages = {152--167},
 title = {{N}onlinear analysis of the {E}ckhaus instability: modulated amplitude waves and phase chaos with nonzero average phase gradient},
 volume = {174},
 year = {2003}
}

@unpublished{budanur2015invariant,
 author = {Budanur, N. B.},
 note = {In preparation},
 title = {{I}nvariant measurement of torsion within eigenframes of periodic orbits},
 year = {2015}
}

@article{budanur2015periodic,
 archiveprefix = {arXiv},
 author = {Budanur, N. B. and Borrero-Echeverry, D. and Cvitanovi{\' c}, P.},
 doi = {10.1063/1.4923742},
 eprint = {1411.3303},
 journal = {Chaos},
 pages = {073112},
 title = {{P}eriodic orbit analysis of a system with continuous symmetry -- {A} tutorial},
 volume = {25},
 year = {2015}
}

@article{budanur2015reduction,
 archiveprefix = {arXiv},
 author = {Budanur, N. B. and Cvitanovi{\' c}, P. and Davidchack, R. L. and Siminos, E.},
 doi = {10.1103/PhysRevLett.114.084102},
 eprint = {1405.1096},
 journal = {Phys. Rev. Lett.},
 pages = {084102},
 primaryclass = {nlin.CD},
 title = {{R}eduction of the {S}O(2) symmetry for spatially extended dynamical systems},
 volume = {114},
 year = {2015}
}

@article{budanur2017heteroclinic,
 archiveprefix = {arXiv},
 author = {N. B. Budanur and B. Hof},
 doi = {10.1017/jfm.2017.516},
 eprint = {1703.10484},
 journal = {J. Fluid Mech.},
 pages = {R1},
 primaryclass = {physics.flu-dyn},
 publisher = {Cambridge University Press},
 title = {{H}eteroclinic path to spatially localized chaos in pipe flow},
 volume = {827},
 year = {2017}
}

@article{budanur2017relative,
 archiveprefix = {arXiv},
 author = {Budanur, N. B. and Short, K. Y. and Farazmand, M. and Willis, A. P. and Cvitanovi{\' c}, P.},
 doi = {10.1017/jfm.2017.699},
 eprint = {1705.03720},
 journal = {J. Fluid Mech.},
 pages = {274--301},
 primaryclass = {physics.flu-dyn},
 publisher = {Cambridge University Press},
 title = {{R}elative periodic orbits form the backbone of turbulent pipe flow},
 volume = {833},
 year = {2017}
}

@article{budanur2017unstable,
 archiveprefix = {arXiv},
 author = {Budanur, N. B. and Cvitanovi{\' c}, P.},
 doi = {10.1007/s10955-016-1672-z},
 eprint = {1509.08133},
 journal = {J. Stat. Phys.},
 number = {3},
 pages = {636--655},
 primaryclass = {nlin.CD},
 title = {{U}nstable manifolds of relative periodic orbits in the symmetry-reduced state space of the {K}uramoto-{S}ivashinsky system},
 volume = {167},
 year = {2017}
}

@article{budanur2018complexity,
 archiveprefix = {arXiv},
 author = {N. B. Budanur and B. Hof},
 doi = {10.1103/PhysRevFluids.3.054401},
 eprint = {1802.01918},
 journal = {Phys. Rev. Fluids},
 numpages = {18},
 pages = {054401},
 primaryclass = {physics.flu-dyn},
 publisher = {American Physical Society},
 title = {{C}omplexity of the laminar-turbulent boundary in pipe flow},
 volume = {3},
 year = {2018}
}

@article{budanur2019geometry,
 archiveprefix = {arXiv},
 author = {N. B. Budanur and A. S. Dogra and B. Hof},
 doi = {10.1103/PhysRevFluids.4.102401},
 eprint = {1810.02211},
 issue = {10},
 journal = {Phys. Rev. Fluids},
 month = {Oct},
 numpages = {7},
 pages = {102401},
 primaryclass = {physics.flu-dyn},
 title = {{G}eometry of transient chaos in streamwise-localized pipe flow turbulence},
 volume = {4},
 year = {2019}
}

@article{budanur2019state,
 archiveprefix = {arXiv},
 author = {N. B. Budanur and M. Fleury},
 doi = {10.1063/1.5058279},
 eprint = {1812.09011},
 journal = {Chaos},
 number = {1},
 pages = {013122},
 title = {{S}tate space geometry of the chaotic pilot-wave hydrodynamics},
 volume = {29},
 year = {2019}
}

@article{budanur2020upper,
 author = {N. B. Budanur and E. Marensi and A. P. Willis and B. Hof},
 doi = {10.1103/PhysRevFluids.5.023903},
 issue = {2},
 journal = {Phys. Rev. Fluids},
 month = {Feb},
 numpages = {11},
 pages = {023903},
 title = {{U}pper edge of chaos and the energetics of transition in pipe flow},
 url = {https://link.aps.org/doi/10.1103/PhysRevFluids.5.023903},
 volume = {5},
 year = {2020}
}

@article{budivsic2012applied,
 archiveprefix = {arXiv},
 author = {Budi{\v s}i{\' c}, M. and Mohr, R. M. and Mezi\textbackslash          ,́ I.},
 doi = {10.1063/1.4772195},
 eprint = {1206.3164},
 journal = {Chaos},
 numpages = {33},
 pages = {047510},
 title = {{A}pplied {K}oopmanism},
 volume = {22},
 year = {2012}
}

@article{budivsic2012geometry,
 abstract = {The
method we present in this paper uses trajectory averages of scalar
functions on the state space to: (a) identify invariant sets in the state
space, and (b) to form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the
space of the trajectory averages of a function basis on the state space.
Second, we endow the ergodic quotient with a metric structure that
successfully captures how similar the invariant sets are in the state
space. Finally, we parametrize the ergodic quotient using intrinsic
diffusion modes on it. By segmenting the ergodic quotient based on the
diffusion modes, we extract coherent features in the state space of the
dynamical system. The algorithm is validated by analyzing the
Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam\textquotesingle          s approximation of the transfer
operator and the computation of Lagrangian Coherent Structures.
Furthermore, we explain how the method extends the Poincar{\'e} map analysis
for periodic flows. As a demonstration, we apply the method to a
periodically-driven three-dimensional Hill\textquotesingle          s vortex flow, discovering
unknown coherent structures in its state space.},
 archiveprefix = {arXiv},
 author = {M. Budi{\v s}i{\' c} and I. Mezi\textbackslash          },
 doi = {10.1016/j.physd.2012.04.006},
 eprint = {1204.2050},
 journal = {Physica D},
 pages = {1255--1269},
 title = {{G}eometry of the ergodic quotient reveals coherent structures in flows},
 volume = {241},
 year = {2012}
}

@article{burke2006localized,
 author = {Burke, J. and Knobloch, E.},
 doi = {10.1103/PhysRevE.73.056211},
 journal = {Phys. Rev. E},
 pages = {056211},
 title = {{L}ocalized states in the generalized {S}wift-{H}ohenberg equation},
 volume = {73},
 year = {2006}
}

@article{burke2007homoclinic,
 author = {Burke, J. and Knobloch, E.},
 doi = {10.1063/1.2746816},
 journal = {Chaos},
 pages = {037102},
 title = {{H}omoclinic snaking: structure and stability},
 volume = {17},
 year = {2007}
}

@article{burke2009swifthohenberg,
 author = {Burke, J. and Houghton, S. M. and Knobloch, E.},
 doi = {10.1103/PhysRevE.80.036202},
 journal = {Phys. Rev. E},
 pages = {036202},
 title = {{S}wift-{H}ohenberg equation with broken reflection symmetry},
 volume = {80},
 year = {2009}
}

@article{bush2015pilotwave,
 author = {J. W.M. Bush},
 doi = {10.1146/annurev-fluid-010814-014506},
 journal = {Annu. Rev. Fluid Mech.},
 number = {1},
 pages = {269--292},
 title = {{P}ilot-{W}ave {H}ydrodynamics},
 url = {
https://doi.org/10.1146/annurev-fluid-010814-014506},
 volume = {47},
 year = {2015}
}

@inproceedings{busse1996bifurcation,
 author = {Busse, F. H. and Clever, R. M.},
 booktitle = {Advances in Turbulence VI},
 editor = {Gavrilakis, S. and Machiels, L. and Monkewitz, P. A.},
 pages = {309--312},
 series = {Proceedings of the Sixth European Turbulence Conference},
 title = {{B}ifurcation sequences in fluid flows and coherent structures in the turbulent state},
 year = {1996}
}

@inbook{busse1996waves,
 address = {New York},
 author = {F. H. Busse and R. M. Clever},
 chapter = {Bifurcation sequences in problems of thermal convection and of plane Couette flow},
 pages = {209--226},
 publisher = {Springer},
 title = {{W}aves and {N}onlinear {P}rocesses in {H}ydrodynamics},
 year = {1996}
}

@article{busse2004visualizing,
 author = {F. H. Busse},
 journal = {Science},
 pages = {1574--1575},
 title = {{V}isualizing the dynamics of the onset of turbulence},
 volume = {305},
 year = {2004}
}

@article{butler1993optimal,
 author = {Butler, K. M. and Farrell, B. F.},
 journal = {Phys. Fluids A},
 number = {3},
 pages = {774--777},
 publisher = {American Institute of Physics},
 title = {{O}ptimal perturbations and streak spacing in wall-bounded turbulent shear flow},
 volume = {5},
 year = {1993}
}

@article{buzug1992fractal,
 author = {Th. Buzug and J. von Stamm and G. Pfister},
 doi = {10.1016/0378-4371(92)90583-C},
 journal = {Physica A},
 pages = {559--563},
 title = {{F}ractal dimensions of strange attractors obtained from the {T}aylor-{C}ouette experiment},
 volume = {191},
 year = {1992}
}

@inproceedings{byrne2014exact,
 author = {G. Byrne and C. D. Marcotte and R. O. Grigoriev},
 booktitle = {Proc. of 8th Conference of the European Study Group on Cardiovascular Oscillations, Trento, Italy},
 publisher = {IEEE},
 title = {{E}xact coherent structures and dynamics of cardiac tissue},
 year = {2014}
}

@article{caldas2019cooperative,
 author = {P. Caldas and M. L{\'o}pez-Pelegr\textbackslash          \textbackslash         'n and D. J. G. Pearce and N. B. Budanur and J. Brugu\textbackslash          \textquotesingle           ́and M. Loose},
 doi = {10.1038/s41467-019-13702-4},
 journal = {Nat. Commun.},
 pages = {5744},
 title = {{C}ooperative ordering of treadmilling filaments in cytoskeletal networks of {F}ts{Z} and it crosslinker {Z}ap{A}},
 volume = {10},
 year = {2019}
}

@book{camacho2004model,
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 year = {2004}
}

@article{cantwell1992exact,
 author = {B. J. Cantwell},
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 journal = {Phys. Fluids A},
 keywords = {VELOCITY; TENSORS; ANALYTICAL SOLUTION; PRESSURE GRADIENTS; STRAIN RATE; TURBULENT FLOW; REYNOLDS NUMBER; NAVIERSTOKES EQUATIONS; KINETIC ENERGY; INCOMPRESSIBLE FLOW; NUMERICAL SOLUTION; ISOTROPY; INITIAL CONDITIONS},
 pages = {782--793},
 title = {{E}xact solution of a restricted {E}uler equation for the velocity gradient tensor},
 volume = {4},
 year = {1992}
}

@book{canuto1988spectral,
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 author = {C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang},
 publisher = {Springer},
 title = {{S}pectral {M}ethods in {F}luid {D}ynamics},
 year = {1988}
}

@book{canuto2006spectral,
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 author = {C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang},
 publisher = {Springer},
 title = {{S}pectral {M}ethods: {F}undamentals in {S}ingle {D}omains},
 year = {2006}
}

@book{canuto2007spectral,
 address = {New York},
 author = {C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang},
 doi = {10.1007/978-3-540-30728-0},
 publisher = {Springer},
 title = {{S}pectral {M}ethods: {E}volution to {C}omplex {G}eometries and {A}pplications to {F}luid {D}ynamics},
 year = {2007}
}

@article{cariello1989painleve,
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 pages = {77},
 title = {{P}ainlev{\'e} expansions for nonintegrable evolution equations},
 volume = {39},
 year = {1989}
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@article{carlson1962inertia,
 author = {D. Carlson and H. Schneider},
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 pages = {479--484},
 title = {{I}nertia theorems for matrices: the semi-definite case},
 volume = {68},
 year = {1962}
}

@article{carlsson2009topology,
 author = {G. Carlsson},
 doi = {10.1090/S0273-0979-09-01249-X},
 issn = {0273-0979},
 journal = {Bull. Amer. Math. Soc.},
 number = {2},
 pages = {255--308},
 title = {{T}opology and data},
 url = {https://doi.org/10.1090/S0273-0979-09-01249-X},
 volume = {46},
 year = {2009}
}

@article{carmona2015noose,
 author = {Carmona, V. and Fern{\'a}ndez-S{\'a}nchez, F. and Garc{\i}́a-Medina, E. and Teruel, A. E.},
 doi = {10.1007/s00332-015-9251-z},
 journal = {J. Nonlin. Sci},
 pages = {1209--1224},
 title = {{N}oose structure and bifurcations of periodic orbits in reversible three-dimensional piecewise linear differential systems},
 volume = {25},
 year = {2015}
}

@article{carroll2013proposed,
 author = {P. L. Carroll and G. Blanquart},
 doi = {http://dx.doi.org/10.1063/1.4826315},
 eid = {105114},
 journal = {Phys. Fluids},
 number = {10},
 title = {{A} proposed modification to {L}undgren\textquotesingle          s physical space velocity forcing method for isotropic turbulence},
 volume = {25},
 year = {2013}
}

@book{cartan1935la,
 address = {Paris},
 author = {Cartan, {\'E}.},
 publisher = {Hermann},
 series = {Expos{\'e}s de G\textbackslash          {\'o}m\textbackslash          \textquotesingle          {\'r}ie},
 title = {{L}a m{\'e}thode du rep{\`e}re mobile, la th\textbackslash          {\'o}rie des groupes continus, et les espaces g\textbackslash          \textquotesingle          \textbackslash          ́\textquotesingle          e{\'l}is\textbackslash          \textquotesingle          es},
 volume = {5},
 year = {1935}
}

@article{carter2004covariant,
 archiveprefix = {arXiv},
 author = {Carter, B. and Chamel, N.},
 eprint = {astro-ph0305186},
 journal = {Int. J. Mod. Phys. D},
 title = {{C}ovariant analysis of {N}ewtonian multi-fluid models for neutron stars {I}: {M}ilne-{C}artan structure and variational formulation},
 year = {2004}
}

@article{cendra1998maxwellvlasov,
 author = {Cendra, H. and Holm, D. D. and Hoyle, M. J. W. and Marsden, J. E.},
 journal = {J. of Math. Phys.},
 pages = {3138--3157},
 title = {{T}he {M}axwell-{V}lasov equations in {E}uler-{P}oincar{\'e} form},
 volume = {39},
 year = {1998}
}

@article{champneys1998homoclinic,
 author = {Champneys, A. R.},
 doi = {10.1016/S0167-2789(97)00209-1},
 journal = {Physica D},
 pages = {158--186},
 title = {{H}omoclinic orbits in reversible systems and their applications in mechanics, fluids and optics},
 volume = {112},
 year = {1998}
}

@article{champneys2008unfolding,
 author = {Champneys, A. R. and Kirk, V. and Knobloch, E. and Oldeman, B. E. and Rademacher, J. D. M.},
 doi = {10.1137/080734923},
 journal = {SIAM J. Appl. Dyn. Syst.},
 pages = {1261--1304},
 title = {{U}nfolding a tangent equilibrium-to-periodic heteroclinic cycle},
 volume = {8},
 year = {2008}
}

@article{chan2007realtime,
 author = {K.-Y. Chan and D. Stich and G. A. Voth},
 journal = {Rev. Sci. Inst.},
 pages = {02374},
 title = {{R}eal-time image compression for high-speed particle tracking},
 volume = {78},
 year = {2007}
}

@article{chandler2013invariant,
 archiveprefix = {arXiv},
 author = {Chandler, G. J. and Kerswell, R. R.},
 bdsk-url-1 = {http://dx.doi.org/10.1017/jfm.2013.122},
 doi = {10.1017/jfm.2013.122},
 eprint = {1207.4682},
 journal = {J. Fluid M.},
 pages = {554--595},
 title = {{I}nvariant recurrent solutions embedded in a turbulent two-dimensional {K}olmogorov flow},
 volume = {722},
 year = {2013}
}

@article{chandre2003timefrequency,
 abstract = {Use wavelets to extract local frequencies and use the
major frequencies to identify the characteristics of phase space
motion.},
 author = {C. Chandre and S. Wiggins and T. Uzer},
 journal = {Physica D},
 pages = {171},
 title = {{T}ime-frequency analysis of chaotic systems},
 volume = {181},
 year = {2003}
}

@article{chang1981transitions,
 author = {Chang, S.-J. and Wright, J.},
 bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevA.23.1419},
 doi = {10.1103/PhysRevA.23.1419},
 journal = {Phys. Rev. A},
 pages = {1419--1433},
 title = {{T}ransitions and distribution functions for chaotic systems},
 volume = {23},
 year = {1981}
}

@article{chang1986travelling,
 abstract = {The lowest order normal form analysis is give to the KSe and the approximate analytical wavelength-amplitude and wavespeed-amplitude relation are given. They agree quite well with the numerical calculations.},
 author = {H.-C. Chang},
 journal = {Phys. Fluids},
 pages = {3142},
 title = {{T}ravelling waves on fluid interfaces: {N}ormal form analysis of the {K}uramoto-{S}ivashisky equation},
 volume = {29},
 year = {1986}
}

@article{chang1994wave,
 author = {H.-C. Chang},
 journal = {Ann. Rev. Fluid Mech.},
 pages = {103--136},
 title = {{W}ave evolution on a falling film},
 volume = {26},
 year = {1994}
}

@article{chantry2014genesis,
 author = {Chantry, M. and Willis, A. P. and Kerswell, R. R.},
 doi = {10.1103/PhysRevLett.112.164501},
 journal = {Phys. Rev. Lett.},
 pages = {164501},
 title = {{G}enesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow},
 volume = {112},
 year = {2014}
}

@article{chantry2014studying,
 author = {Chantry, M. and Schneider, T. M.},
 journal = {J. Fluid Mech.},
 pages = {506--517},
 title = {{S}tudying edge geometry in transiently turbulent shear flows},
 volume = {747},
 year = {2014}
}

@article{chantry2016turbulentlaminar,
 author = {M. Chantry and L. S. Tuckerman and D. Barkley},
 doi = {10.1017/jfm.2016.92},
 journal = {J. Fluid Mech.},
 pages = {R8},
 title = {{T}urbulent-laminar patterns in shear flows without walls},
 volume = {791},
 year = {2016}
}

@article{chantry2017universal,
 author = {M. Chantry and L. S. Tuckerman and D. Barkley},
 doi = {10.1017/jfm.2017.405},
 journal = {J. Fluid Mech.},
 pages = {R1},
 title = {{U}niversal continuous transition to turbulence in a planar shear flow},
 volume = {824},
 year = {2017}
}

@article{chapman2002subcritical,
 author = {S. J. Chapman},
 journal = {J. Fluid Mech.},
 pages = {35--97},
 title = {{S}ubcritical transition in channel flows},
 volume = {451},
 year = {2002}
}

@article{chate1987transition,
 author = {H. Chat{\'e} and P. Manneville},
 journal = {Phys. Rev. Lett.},
 pages = {112},
 title = {{T}ransition to {T}urbulence via spatiotemporal {I}ntermittency},
 volume = {58},
 year = {1987}
}

@article{chate1994spatiotemporal,
 abstract = {In the Benjamin-Feir stable region, spatiotemporal
intermittency regimes are identified which consists of patches of
linearly stale stable plane waves separated by localized objects with
a well defined dynamics. In the transition reginon, asymptotic states
with an irregular, frozen spatial structure are shown to occur. The
bistable region beyond the Benjamin-Feir stable region is of the same
spatiotemporal intermittency type with phase turbulence as the },
 author = {H. Chat{\'e}},
 journal = {Nonlinearity},
 pages = {185--204},
 title = {{S}patiotemporal intermittency regimes of the one-dimensional complex {G}inzburg-{L}andau equation},
 volume = {7},
 year = {1994}
}

@book{chen1989group,
 address = {Singapore},
 author = {Chen, J.-Q. and Ping, J. and Wang, F.},
 doi = {10.1142/0262},
 publisher = {World Scientific},
 title = {{G}roup {R}epresentation {T}heory for {P}hysicists},
 year = {1989}
}

@article{cheng1990existence,
 author = {Cheng, C. Q. and Sun, Y. S.},
 journal = {Celestial Mech. Dynam. Astronom.},
 pages = {275--92},
 title = {{E}xistence of invariant tori in three-dimensional measure-preserving mappings},
 volume = {47},
 year = {1990}
}

@article{cherhabili1995spatially,
 author = {A. Cherhabili and U. Ehrenstein},
 journal = {Eur. J. Mech. B Fluids},
 pages = {677--696},
 title = {{S}patially localized two-dimensional finite-amplitude states in plane {C}ouette flow},
 url = {http://cat.inist.fr/?aModele=afficheN\&cpsidt=2903887},
 volume = {14},
 year = {1995}
}

@article{cherhabili1996existence,
 author = {A. Cherhabili and U. Ehrenstein},
 doi = {10.1007/978-94-009-0297-8\_89},
 journal = {Fluid Mechanics and its Applications},
 pages = {317--320},
 title = {{E}xistence and stability of finite-amplitude states in plane {C}ouette flow},
 volume = {36},
 year = {1996}
}

@article{cherhabili1997finiteamplitude,
 author = {Cherhabili, A. and Ehrenstein, U.},
 journal = {J. Fluid Mech.},
 pages = {159--177},
 title = {{F}inite-amplitude equilibrium states in plane {C}ouette flow},
 volume = {342},
 year = {1997}
}

@article{cherhabili2000nonlinear,
 author = {A. Cherhabili},
 issn = {0179-9614},
 journal = {Notes in Numerical Fluid Mechanics},
 pages = {69--80},
 title = {{N}onlinear equilibrium states in plane {C}ouette flow: {C}ontinuation from {P}oiseuille flow to {C}ouette flow},
 volume = {74},
 year = {2000}
}

@article{cherry2017introduction,
 author = {E. M. Cherry  and F. H. Fenton  and T. Krogh-Madsen  and S. Luther and U. Parlitz},
 doi = {10.1063/1.5003940},
 journal = {Chaos},
 number = {9},
 pages = {093701},
 title = {{I}ntroduction to {F}ocus {I}ssue: {C}omplex {C}ardiac {D}ynamics},
 volume = {27},
 year = {2017}
}

@article{chertkov1999lagrangian,
 archiveprefix = {arXiv},
 author = {Chertkov, M. and Pumir, A. and Shraiman, B. I.},
 eprint = {chao-dyn9905027},
 journal = {Phys. Fluids},
 title = {{L}agrangian tetrad dynamics and the phenomenology of turbulence},
 year = {1999}
}

@article{cherubini2014minimal,
 author = {S. Cherubini and P. De Palma},
 journal = {Fluid Dyn. Res},
 number = {4},
 pages = {041403},
 title = {{M}inimal perturbations approaching the edge of chaos in a {C}ouette flow},
 url = {http://stacks.iop.org/1873-7005/46/i=4/a=041403},
 volume = {46},
 year = {2014}
}

@article{chillingworth2000generic,
 author = {D. Chillingworth},
 journal = {Dyn. Stab. Syst.},
 pages = {101--137},
 title = {{G}eneric multiparameter bifurcation from a manifold},
 volume = {15},
 year = {2000}
}

@article{choi1994active,
 author = {H. Choi and P. Moin and J. Kim},
 journal = {J. Fluid Mech.},
 pages = {75},
 title = {{A}ctive turbulence control for drag reduction in wall-bounded flows},
 volume = {262},
 year = {1994}
}

@article{choi2014on,
 author = {M. Choi and T. P. Sapsis and G. E. Karniadakis},
 doi = {10.1016/j.jcp.2014.03.050},
 journal = {J. Comput. Physics},
 pages = {1--20},
 title = {{O}n the equivalence of dynamically orthogonal and bi-orthogonal methods: {T}heory and numerical simulations},
 volume = {270},
 year = {2014}
}

@article{chossat1988symmetryincreasing,
 author = {P. Chossat and M. Golubitsky},
 doi = {10.1016/0167-2789(88)90066-8},
 journal = {Physica D},
 number = {3},
 pages = {423 -- 436},
 title = {{S}ymmetry-increasing bifurcation of chaotic attractors},
 volume = {32},
 year = {1988}
}

@book{chossat1994taylorcouette,
 address = {New York},
 author = {P. Chossat and G. Iooss},
 publisher = {Springer},
 title = {{T}he {T}aylor-{C}ouette {P}roblem},
 year = {1994}
}

@book{chossat2000methods,
 address = {Singapore},
 author = {P. Chossat and R. Lauterbach},
 isbn = {978-981-02-3828-5},
 publisher = {World Scientific},
 title = {{M}ethods in {E}quivariant {B}ifurcations and {D}ynamical {S}ystems},
 year = {2000}
}

@unpublished{choueiri2020crises,
 author = {G. Choueiri and B. Suri and J. Merrin and M. Serbyn and B. Hof and N. B. Budanur},
 note = {In preparation},
 title = {{C}rises and chaotic scattering in hydrodynamic pilot-wave experiments},
 year = {2020}
}

@article{christiansen1995generating,
 archiveprefix = {arXiv},
 author = {F. Christiansen and A. Politi},
 eprint = {chao-dyn9411005},
 journal = {Phys. Rev. E},
 title = {{A} generating partition for the standard map},
 year = {1995}
}

@article{christiansen1996symbolic,
 author = {F. Christiansen and A. Politi},
 journal = {Nonlinearity},
 pages = {1623--1640},
 title = {{S}ymbolic encoding in symplectic maps},
 volume = {9},
 year = {1996}
}

@article{christiansen1997guidelines,
 author = {F. Christiansen and A. Politi},
 journal = {Physica D},
 pages = {32--41},
 title = {{G}uidelines for the construction of a generating partition in the standard map},
 volume = {109},
 year = {1997}
}

@article{christiansen1997spatiotemporal,
 archiveprefix = {arXiv},
 author = {F. Christiansen and P. Cvitanovi{\' c} and V. Putkaradze},
 doi = {10.1088/0951-7715/10/1/004},
 eprint = {chao-dyn9606016},
 journal = {Nonlinearity},
 pages = {55--70},
 title = {{S}patiotemporal chaos in terms of unstable recurrent patterns},
 volume = {10},
 year = {1997}
}

@article{chumakov2007scaling,
 author = {S. G. Chumakov},
 doi = {10.1063/1.2735001},
 journal = {Phys. Fluids},
 number = {5},
 pages = {058104},
 title = {{S}caling properties of subgrid-scale energy dissipation},
 volume = {19},
 year = {2007}
}

@article{chumakov2008priori,
 author = {S. G. Chumakov},
 doi = {10.1103/PhysRevE.78.036313},
 issue = {3},
 journal = {Phys. Rev. E},
 numpages = {11},
 pages = {036313},
 title = {{A} priori study of subgrid-scale flux of a passive scalar in isotropic homogeneous turbulence},
 volume = {78},
 year = {2008}
}

@article{churchland2012neural,
 author = {Churchland, M. M. and Cunningham, J. P. and Kaufman, M. T. and Foster, J. D. and Nuyujukian, P. and Ryu, S. I. and Shenoy, K. V.},
 doi = {10.1038/nature11129},
 journal = {Nature},
 pages = {51--56},
 title = {{N}eural population dynamics during reaching},
 volume = {487},
 year = {2012}
}

@article{ciro2018efficient,
 archiveprefix = {arXiv},
 author = {D. Ciro and I.\textasciitilde          L. Caldas and  R.\textasciitilde          L. Viana and T.\textasciitilde          E. Evans},
 eprint = {1710.10140},
 title = {{E}fficient manifold tracing for planar maps},
 year = {2018}
}

@misc{civile2020dati,
 author = {Dipartimento della Protezione Civile},
 title = {{D}ati {C}OV{I}D-19 {I}talia},
 url = {https://github.com/pcm-dpc/COVID-19},
 year = {2020}
}

@article{clever1974large,
 author = {R. Clever and F. Busse},
 journal = {J. Fluid Mech.},
 keywords = {fluid dynamics},
 pages = {665},
 title = {{L}arge wavelength convection rolls in low {P}randt number fluids},
 volume = {65},
 year = {1974}
}

@article{clever1992threedimensional,
 author = {Clever, R. M. and Busse, F. H.},
 journal = {J. Fluid Mech.},
 pages = {511--527},
 title = {{T}hree-dimensional convection in a horizontal layer subjected to constant shear},
 volume = {234},
 year = {1992}
}

@article{clever1997tertiary,
 abstract = {The manifold of Nagata steady solutions is
explored in the parameter space of the problem and their
instabilities are investigated. These instabilities usually
lead to time-periodic solutions whose properties do not
differ much from those of the steady solutions except that
the amplitude varies in time. In some cases travelling wave
solutions which are asymmetric with respect to the midplane
of the layer are found as quaternary states of flow.
Similarities with longitudinal vortices recently observed in
experiments are discussed.},
 author = {Clever, R. M. and Busse, F. H.},
 journal = {J. Fluid Mech.},
 pages = {137--153},
 title = {{T}ertiary and quaternary solutions for plane {C}ouette flow},
 volume = {344},
 year = {1997}
}

@incollection{clever2000tertiary,
 author = {Clever, R. M. and Busse, F. H.},
 booktitle = {Physics of Rotating Fluids},
 doi = {10.1007/3-540-45549-3\_22},
 editor = {Egbers, C. and Pfister, G.},
 isbn = {978-3-540-67514-3},
 pages = {399--416},
 publisher = {Springer},
 title = {{T}ertiary and quaternary solutions for plane {C}ouette flow with thermal stratification},
 year = {2000}
}

@article{cliffe1983numerical,
 author = {K. A. Cliffe},
 doi = {10.1017/S0022112083003055},
 journal = {J. Fluid Mech.},
 pages = {219--233},
 title = {{N}umerical calculations of two-cell and single-cell {T}aylor flows},
 volume = {135},
 year = {1983}
}

@phdthesis{coffee1976stability,
 author = {Coffee, T. P.},
 school = {City Univ. of New York, NY.},
 title = {{T}he stability of plane {C}ouette flow},
 year = {1976}
}

@book{coffey2012langevin,
 author = {W. T. Coffey and Y. P. Kalmykov},
 doi = {10.1142/8195},
 edition = {3rd},
 publisher = {WORLD SCIENTIFIC},
 title = {{T}he {L}angevin {E}quation},
 year = {2012}
}

@article{cohen1972nonlinear,
 author = {B. I. Cohen and J. A. Krommes and W. M. Tang and M. N. Rosenbluth},
 journal = {Nucl. Fusion},
 pages = {971--991},
 title = {{N}on-linear saturation of the dissipative trapped-ion mode by mode coupling},
 volume = {16},
 year = {1972}
}

@article{cohensteiner2007stability,
 author = {D. Cohen-Steiner and H. Edelsbrunner and J. Harer},
 doi = {10.1007/s00454-006-1276-5},
 journal = {Discrete Comput Geom.},
 number = {1},
 pages = {103--120},
 title = {{S}tability of {P}ersistence {D}iagrams},
 volume = {37},
 year = {2007}
}

@book{collet1980iterated,
 address = {Boston},
 author = {P. Collet and J.-P. Eckmann},
 publisher = {Birkh{\"a}user},
 title = {{I}terated {M}aps on the {I}nterval as {D}ynamical {S}ystems},
 year = {1980}
}

@article{collet1980universal,
 author = {P. Collet and J.-P. Eckmann and O. E. Lanford},
 journal = {Commun. Math. Phys.},
 pages = {211},
 title = {{U}niversal {P}roperties of {M}aps on an interval},
 volume = {76},
 year = {1980}
}

@article{collet1993analyticity,
 abstract = {Proved the analyticity of the solution of the KSe on a
periodic interval near the real axis and the width of the region is
estimated. Numerical calculations show that this width should not
depend on the size of the interval.},
 author = {P. Collet and J.-P. Eckmann and H. Epstein and J. Stubbe},
 journal = {Physica D},
 pages = {321--326},
 title = {{A}nalyticity for the {K}uramoto-{S}ivashinsky equation},
 volume = {67},
 year = {1993}
}

@article{collet1993global,
 abstract = {New bounds are given for the L\^{}2 norm of solutions of the KSe (not necessarily antisymmetric).},
 author = {P. Collet and J.-P. Eckmann and H. Epstein and J. Stubbe},
 journal = {Commun. Math. Phys.},
 pages = {203--214},
 title = {{A} global attracting set for the {K}uramoto-{S}ivashinsky equation},
 volume = {152},
 year = {1993}
}

@article{collis2004issues,
 author = {S. S. Collis and R. D. Joslin and A. Seifert and V. Theofilis},
 journal = {Prog. Aerosp. Sci.},
 pages = {237},
 title = {{I}ssues in active flow control: {T}heory, control, simulation, and experiment},
 volume = {40},
 year = {2004}
}

@book{constantin1989integral,
 address = {New York},
 author = {P. Constantin and C. Foias and B. Nicolaenko and R. Temam},
 publisher = {Springer},
 title = {{I}ntegral {M}anifolds and {I}nertial {M}anifolds for {D}issipative {P}artial {D}ifferential {E}quations},
 year = {1989}
}

@book{conte1999painleve,
 address = {New York},
 author = {R. Conte},
 publisher = {Springer},
 title = {{T}he {P}ainlev{\'e} property: one century later},
 year = {1999}
}

@book{cornwell1997group,
 address = {New York},
 author = {Cornwell, J. F.},
 publisher = {Academic Press},
 title = {{G}roup {T}heory in {P}hysics: {A}n introduction},
 year = {1997}
}

@article{couder2005from,
 author = {Y. Couder and E. Fort and C.-H. Gautier and A. Boudaoud},
 journal = {Phys. Rev. Lett.},
 number = {17},
 pages = {177801},
 publisher = {American Physical Society},
 title = {{F}rom bouncing to floating: {N}oncoalescence of drops on a fluid bath},
 volume = {94},
 year = {2005}
}

@article{couder2005walking,
 author = {Y. Couder and S. Proti{\`e}re and E. Fort and A. Boudaoud},
 journal = {Nature},
 number = {7056},
 pages = {208},
 publisher = {Nature Publishing Group},
 title = {{W}alking and orbiting droplets},
 volume = {437},
 year = {2005}
}

@article{couder2006singleparticle,
 author = {Y. Couder and E. Fort},
 journal = {Phys. Rev. Lett.},
 number = {15},
 pages = {154101},
 publisher = {APS},
 title = {{S}ingle-particle diffraction and interference at a macroscopic scale},
 volume = {97},
 year = {2006}
}

@article{coullet1978iterations,
 author = {P. Coullet and C. Tresser},
 journal = {J. de Physique Colloque C},
 pages = {5--25},
 title = {{I}t{\'e}rations d\textquotesingle          endomorphismes et groupe de renormalisation},
 volume = {39},
 year = {1978}
}

@article{coullet1978iterationsb,
 author = {P. Coullet and C. Tresser},
 journal = {C. R. Acad. Sci. Paris A},
 pages = {577--580},
 title = {{I}t{\'e}rations d\textquotesingle          endomorphismes et groupe de renormalisation},
 volume = {287},
 year = {1978}
}

@book{cox1996ideals,
 address = {New York},
 author = {D. A. Cox and J. B. Little and D. 0\textquotesingle          Shea},
 publisher = {Springer},
 title = {{I}deals, {V}arieties and {A}lgorithms},
 year = {1996}
}

@article{craik1972eigenvalue,
 abstract = {New eigenvalue bounds are derived for the linear
stability of inviscid parallel flows, both for homogeneous and for
stratified fluids. The usefulness of these bounds, as compared with
that of previous results, is assessed for several examples. For
homogeneous fluids the new upper bounds for the imaginary part ci of
the complex phase velocity are sometimes better than previous
criteria. For both homogeneous and stratified flows, the new upper
bounds for the wave-number ? of neutrally stable disturbances
improve on previous results, giving values within 10\% of the known
exact solution in several cases.},
 author = {A. D. D. Craik},
 journal = {J. Fluid Mech.},
 pages = {657--670},
 title = {{E}igenvalue bounds in linear inviscid stability theory},
 volume = {53},
 year = {1972}
}

@article{craik1989stability,
 abstract = {A formulation, previously employed to find exact
Navier-Stokes solutions for planar disturbances in two- and
three-dimensional flows with spatially uniform rates of strain, is here
adapted to incorporate the contribution of various types of body force.
In the absence of body forces, it is known that unbounded flows with
constant vorticity and elliptical streamlines are unstable to certain
planar disturbances, which are amplified by a Floquet mechanism. The
influence of a Coriolis force upon this instability mechanism is here
described in detail, as an illustration of the general formulation. The
results are likely to be of geophysical interest and may also have
relevance to the breakdown of closed-eddy structures in turbulence. The
final section of the paper reviews other systems for which analogous
exact solutions may be obtained.},
 author = {A. D. D. Craik},
 journal = {J. Fluid Mech.},
 pages = {275--292},
 title = {{T}he stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions},
 volume = {198},
 year = {1989}
}

@misc{cresson2006hyperboliity,
 archiveprefix = {arXiv},
 author = {J. Cresson and C. Guillet},
 eprint = {math/0601701},
 title = {{H}yperboliity versus partial-hyperbolicity and the transversality-torsion phenomenon},
 year = {2006}
}

@book{creutz1992quantum,
 address = {Singapore},
 author = {M. Creutz},
 publisher = {World Scientific},
 title = {{Q}uantum {F}ields on the {C}omputer},
 year = {1992}
}

@article{crisanti1993intermittency,
 author = {A. Crisanti and M. H. Jensen and A. Vulpiani and G. Paladin},
 doi = {10.1103/physrevlett.70.166},
 journal = {Phys. Rev. Lett.},
 month = {jan},
 number = {2},
 pages = {166--169},
 publisher = {American Physical Society (APS)},
 title = {{I}ntermittency and predictability in turbulence},
 url = {https://doi.org/10.1103\%2Fphysrevlett.70.166},
 volume = {70},
 year = {1993}
}

@article{cristeaplaton2018walking,
 author = {T. Cristea-Platon and P. J. S{\'a}enz and J. W. M. Bush},
 doi = {10.1063/1.5034123},
 journal = {Chaos},
 number = {9},
 pages = {096116},
 title = {{W}alking droplets in a circular corral: {Q}uantisation and chaos},
 volume = {28},
 year = {2018}
}

@article{crofts2006efficient,
 author = {J. J. Crofts and R. L. Davidchack},
 journal = {SIAM J. Sci. Comp.},
 pages = {1275--1288},
 title = {{E}fficient detection of periodic orbits in chaotic systems by stabilizing transformations},
 volume = {28},
 year = {2006}
}

@phdthesis{crofts2007efficient,
 address = {Leicester, UK},
 archiveprefix = {arXiv},
 author = {J. J. Crofts},
 eprint = {nlin.CD0706.1940},
 school = {Univ. of Leicester},
 title = {{E}fficient method for detection of periodic orbits in chaotic maps and flows},
 year = {2007}
}

@article{cross1993pattern,
 author = {M. C. Cross and P. C. Hohenberg},
 journal = {Rev. Mod. Phys},
 title = {{P}attern formation outside of equilibrium},
 volume = {65},
 year = {1993}
}

@article{cross1994spatiotemporal,
 abstract = {The phenomena of spatiotermporal chaos are introduced based on both rotating and regular Rayleigh-Bernard convection. The enormous number of available degrees of freedom is a vivid feature. A set of interesting questions have been raised, including how to characterize a spatiotemporal chaotic state. The importance of the study of small subsystems is emphasized.},
 author = {M. C. Cross and P. C. Hohenberg},
 journal = {Science},
 pages = {1569--1570},
 title = {{S}patiotemporal {C}haos},
 volume = {263},
 year = {1994}
}

@conference{crowley2015experimental,
 author = {C. Crowley and M. Krygier and D. Borrero-Echeverry and R. Grigoriev and M. Schatz},
 booktitle = {68\textbackslash          textsuperscriptth Annual Meeting of the APS Division of Fluid Dynamics},
 series = {Bull. Am. Phys. Soc},
 title = {{E}xperimental observations of direct laminar-turbulent transition in counter-rotating {T}aylor-{C}ouette flow},
 url = {http://meetings.aps.org/link/BAPS.2015.DFD.D29.2},
 volume = {60},
 year = {2015}
}

@article{csordas2005new,
 author = {G. Csordas and M. Charalambides and F. Waleffe},
 journal = {Proc. Amer. Math. Soc.},
 pages = {3551--3560},
 title = {{A} new property of a class of {J}acobi polynomials},
 volume = {133},
 year = {2005}
}

@article{cvitanovic1976group,
 author = {Cvitanovi{\' c}, P.},
 bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevD.14.1536},
 doi = {10.1103/PhysRevD.14.1536},
 journal = {Phys. Rev. D},
 pages = {1536--1553},
 title = {{G}roup theory for {F}eynman diagrams in non-{A}belian gauge theories},
 volume = {14},
 year = {1976}
}

@misc{cvitanovic1977classical,
 author = {P. Cvitanovi{\' c}},
 bdsk-url-1 = {www.nbi.dk/GroupTheory},
 note = {(Oxford University preprint 40/77, unpublished)},
 title = {{C}lassical and exceptional {L}ie algebras as invariance algebras},
 url = {www.nbi.dk/GroupTheory},
 year = {1977}
}

@article{cvitanovic1985scaling,
 author = {P. Cvitanovi{\' c} and B. Shraiman and B. S{\"o}derberg},
 bdsk-url-1 = {http://dx.doi.org/10.1088/0031-8949/32/4/003},
 doi = {10.1088/0031-8949/32/4/003},
 journal = {Phys. Scr.},
 pages = {263--270},
 title = {{S}caling laws for mode lockings in circle maps},
 volume = {32},
 year = {1985}
}

@article{cvitanovic1988invariant,
 author = {P. Cvitanovi{\' c}},
 doi = {10.1103/PhysRevLett.61.2729},
 journal = {Phys. Rev. Lett.},
 pages = {2729--2732},
 title = {{I}nvariant measurement of strange sets in terms of cycles},
 volume = {61},
 year = {1988}
}

@article{cvitanovic1988topological,
 author = {P. Cvitanovi{\' c} and G. H. Gunaratne and I. Procaccia},
 journal = {Phys. Rev. A},
 pages = {1503},
 title = {{T}opological and metric properties of {H}{\'e}non-type strange attractors},
 volume = {38},
 year = {1988}
}

@article{cvitanovic1989periodic,
 author = {Cvitanovi{\' c}, P. and Eckhardt, B.},
 journal = {Phys. Rev. Lett.},
 pages = {823--826},
 title = {{P}eriodic orbit quantization of chaotic systems},
 volume = {63},
 year = {1989}
}

@book{cvitanovic1989universality,
 address = {Bristol},
 author = {P. Cvitanovi{\' c}},
 publisher = {Adam Hilger},
 title = {{U}niversality in {C}haos},
 year = {1989}
}

@article{cvitanovic1991periodic,
 author = {P. Cvitanovi{\' c} and B. Eckhardt},
 journal = {J. Phys. A},
 pages = {L237},
 title = {{P}eriodic orbit expansions for classical smooth flows},
 volume = {24},
 year = {1991}
}

@article{cvitanovic1991periodicb,
 author = {P. Cvitanovi{\' c}},
 journal = {Physica D},
 pages = {138},
 title = {{P}eriodic orbits as the skeleton of classical and quantum chaos},
 volume = {51},
 year = {1991}
}

@article{cvitanovic1992investigation,
 author = {Cvitanovi{\' c}, P. and Gaspard, P. and Schreiber, T.},
 doi = {10.1063/1.165902},
 journal = {Chaos},
 pages = {85--90},
 title = {{I}nvestigation of the {L}orentz gas in terms of periodic orbits},
 volume = {2},
 year = {1992}
}

@article{cvitanovic1992periodic,
 author = {P. Cvitanovi{\' c}},
 journal = {Chaos},
 number = {1},
 pages = {1--4},
 publisher = {AIP},
 title = {{P}eriodic orbit theory in classical and quantum mechanics},
 volume = {2},
 year = {1992}
}

@article{cvitanovic1993entire,
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and G. Vattay },
 eprint = {chao-dyn9307012},
 journal = {Phys. Rev. Lett.},
 title = {{E}ntire {F}redholm determinants for evaluation of semiclassical and thermodynamical spectra},
 year = {1993}
}

@article{cvitanovic1993symmetry,
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and B. Eckhardt},
 eprint = {chao-dyn9303016},
 journal = {Nonlinearity},
 title = {{S}ymmetry decomposition of chaotic dynamics},
 year = {1993}
}

@article{cvitanovic1995transport,
 archiveprefix = {arXiv},
 author = {Cvitanovi{\' c}, P. and Eckmann, J.-P. and Gaspard, P.},
 eprint = {chao-dyn9312003},
 journal = {Chaos Solit. Fract.},
 title = {{T}ransport properties of the {L}orentz gas in terms of periodic orbits},
 year = {1995}
}

@article{cvitanovic1998trace,
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and C. P. Dettmann and R. Mainieri and G. Vattay},
 eprint = {chao-dyn9807034},
 journal = {J. Stat. Phys.},
 title = {{T}race formulas for stochastic evolution operators: {W}eak noise perturbation theory},
 year = {1998}
}

@article{cvitanovic1999spectrum,
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and N. S{\o}ndergaard and G. Palla and G. Vattay and C. P. Dettmann},
 eprint = {chao-dyn9904027},
 journal = {Phys. Rev. E},
 title = {{S}pectrum of stochastic evolution operators: {L}ocal matrix representation approach},
 year = {1999}
}

@article{cvitanovic1999trace,
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and C. P. Dettmann and R. Mainieri and G. Vattay},
 eprint = {chao-dyn9811003},
 journal = {Nonlinearity},
 title = {{T}race formulae for stochastic evolution operators: {S}mooth conjugation method},
 year = {1999}
}

@article{cvitanovic2000chaotic,
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c}},
 eprint = {nlin.CD0001034},
 journal = {Physica A},
 title = {{C}haotic field theory: {A} sketch},
 year = {2000}
}

@inproceedings{cvitanovic2003turbulent,
 address = {Singapore},
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and Y. Lan},
 booktitle = {Proceedings of 10\textbackslash          textsuperscriptth International Workshop on Multiparticle Production: Correlations and Fluctuations in QCD},
 editor = {N. Antoniou},
 eprint = {nlin.CD0308006},
 title = {{T}urbulent fields and their recurrences},
 year = {2003}
}

@book{cvitanovic2008group,
 address = {Princeton},
 author = {P. Cvitanovi{\' c}},
 note = {\textbackslash          tt birdtracks.eu},
 publisher = {Princeton Univ. Press},
 title = {{G}roup {T}heory - {B}irdtracks, {L}ie\textquotesingle          s, and {E}xceptional {G}roups},
 year = {2008}
}

@misc{cvitanovic2008notitle,
 author = {P. Cvitanovi{\' c}},
 note = {chapter \textbackslash          em \textasciigrave          \textasciigrave          World in a mirror\textquotesingle          \textquotesingle          , in Ref.\textasciitilde          i̧teDasBuch},
 year = {2008}
}

@inproceedings{cvitanovic2009geometry,
 address = {Berlin},
 author = {Cvitanovi{\' c}, P. and Gibson, J. F.},
 booktitle = {Advances in Turbulence XII},
 doi = {10.1007/978-3-642-03085-7-17},
 editor = {B. Eckhardt},
 pages = {75--78},
 publisher = {Springer},
 series = {Proc. 12\textbackslash          textsuperscriptth EUROMECH Eur. Turb. Conf., Marburg},
 title = {{G}eometry of state space in plane {C}ouette flow},
 year = {2009}
}

@article{cvitanovic2009on,
 archiveprefix = {arXiv},
 author = {Cvitanovi{\' c}, P. and Davidchack, R. L. and Siminos, E.},
 eprint = {0709.2944},
 journal = {SIAM J. Appl. Dyn. Syst.},
 pages = {1--33},
 title = {{O}n the state space geometry of the {K}uramoto-{S}ivashinsky flow in a periodic domain},
 volume = {9},
 year = {2009}
}

@article{cvitanovic2010geometry,
 author = {Cvitanovi{\' c}, P. and Gibson, J. F.},
 doi = {10.1088/0031-8949/2010/T142/014007},
 journal = {Phys. Scr.},
 pages = {014007},
 title = {{G}eometry of turbulence in wall-bounded shear flows: {P}eriodic orbits},
 volume = {T142},
 year = {2010}
}

@misc{cvitanovic2011appendix,
 author = {P. Cvitanovi{\' c} and G. Vattay},
 note = {in efrefD{\aa}sBuch},
 title = {{A}ppendix \textasciigrave          \textasciigrave          {A}pplications\textquotesingle          \textquotesingle          },
 year = {2011}
}

@article{cvitanovic2012cartography,
 archiveprefix = {arXiv},
 author = {Cvitanovi{\' c}, P. and Borrero-Echeverry, D. and Carroll, K. and Robbins, B. and Siminos, E.},
 doi = {10.1063/1.4758309},
 eprint = {1209.4915},
 journal = {Chaos},
 pages = {047506},
 title = {{C}artography of high-dimensional flows: {A} visual guide to sections and slices},
 volume = {22},
 year = {2012}
}

@inproceedings{cvitanovic2012knowing,
 address = {Melville, New York},
 archiveprefix = {arXiv},
 author = {P. Cvitanovi{\' c} and D. Lippolis},
 booktitle = {Let\textquotesingle          s Face Chaos through Nonlinear Dynamics},
 doi = {10.1063/1.4745574},
 editor = {M. Robnik and V. G. Romanovski},
 eprint = {1206.5506},
 pages = {82--126},
 publisher = {Am. Inst. of Phys.},
 title = {{K}nowing when to stop: {H}ow noise frees us from determinism},
 year = {2012}
}

@inproceedings{cvitanovic2012revealing,
 author = {Cvitanovi{\' c}, P. and Willis, A. P. and Avila, M.},
 booktitle = {Proceed. ICTAM 2012 Intern. Congr. Theor. and Appl. Mech.},
 editor = {J. Wang},
 title = {{R}evealing the state space of turbulent pipe flow by symmetry reduction},
 year = {2012}
}

@incollection{cvitanovic2013chapter,
 address = {Copenhagen},
 author = {P. Cvitanovi{\' c}},
 booktitle = {Chaos: Classical and Quantum},
 editor = {P. Cvitanovi{\' c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
 note = {ChaosBook.org/version14},
 publisher = {Niels Bohr Inst.},
 title = {{C}hapter \textasciigrave          \textasciigrave          {T}race {F}ormulas\textquotesingle          \textquotesingle          },
 year = {2013}
}

@article{cvitanovic2013recurrent,
 author = {Cvitanovi{\' c}, P.},
 doi = {10.1017/jfm.2013.198},
 journal = {J. Fluid Mech. Focus Fluids},
 pages = {1--4},
 title = {{R}ecurrent flows: the clockwork behind turbulence},
 volume = {726},
 year = {2013}
}

@book{cvitanovic2017chaos,
 address = {Copenhagen},
 author = {P. Cvitanovi{\' c} and R. Artuso and R. Mainieri and G. Tanner and G. Vattay},
 note = {\textbackslash          textttwebbook, stable version 15.9, ChaosBook.org},
 publisher = {Niels Bohr Inst.},
 title = {{C}haos: {C}lassical and {Q}uantum},
 year = {2017}
}

@article{dahlqvist1990existence,
 author = {P. Dahlqvist and G. Russberg},
 journal = {Phys. Rev. Lett.},
 pages = {2837},
 title = {{E}xistence of stable orbits in the \textdollar          x\^{}2y\^{}2\textdollar           potential},
 volume = {65},
 year = {1990}
}

@article{dahlqvist1991cycle,
 author = {P. Dahlqvist and G. Russberg},
 journal = {J. Phys. A},
 pages = {4763},
 title = {{C}ycle expansions for intermittent diffusion},
 volume = {24},
 year = {1991}
}

@article{dahlqvist1991periodic,
 author = {P. Dahlqvist and G. Russberg},
 bdsk-url-1 = {http://dx.doi.org/10.1088/0305-4470/24/20/012},
 doi = {10.1088/0305-4470/24/20/012},
 journal = {J. Phys. A},
 pages = {4763--4778},
 title = {{P}eriodic orbit quantization of bound chaotic systems},
 volume = {24},
 year = {1991}
}

@article{dahlqvist1994determination,
 abstract = {We consider the computation of the eigenvalues of the
evolution operator-the resonance spectrum-by means of the zeros of
a zeta function. In particular we address the problems of applying
this formalism to bound chaotic systems, caused by e.g.
intermittency and non-completeness of the symbolic dynamics. For
bound intermittent systems we derive an approximation of the zeta
function. With the aid of this zeta function it is argued that
bound systems with long time tails have branch cuts in the zeta
function and traces (of the evolution operator) approaching unity
as a power law. We also show that the dominant time scale can be
much longer than the period of the shortest periodic orbit, as is,
for example, the case for the hyperbola billiard. Isolated zeros of
the zeta function for the hyperbola billiard are evaluated by means
of a cycle expansion. Crucial for the success of this approach is
the identification of a sequence of periodic orbit responsible for
a logarithmic branch cut in the zeta function. Semiclassical
implications are briefly discussed.},
 author = {P. Dahlqvist},
 bdsk-url-1 = {http://dx.doi.org/10.1088/0305-4470/27/3/020},
 doi = {10.1088/0305-4470/27/3/020},
 journal = {J. Phys. A},
 pages = {763--785},
 title = {{D}etermination of resonance spectra for bound chaotic systems},
 volume = {27},
 year = {1994}
}

@article{dangelmayr1986steadystate,
 author = {Dangelmayr, G.},
 bdsk-url-1 = {http://dx.doi.org/10.1080/02681118608806011},
 doi = {10.1080/02681118608806011},
 journal = {Dyn. Sys.},
 pages = {159--185},
 title = {{S}teady-state mode interactions in the presence of 0(2)-symmetry},
 volume = {1},
 year = {1986}
}

@article{dankowicz1996local,
 author = {H. Dankowicz and P. Holmes and G. Berkooz and J. Elezgaray},
 journal = {Physica D},
 pages = {387--407},
 title = {{L}ocal models of spatio-temporally complex fields},
 volume = {90},
 year = {1996}
}

@article{dauchot1995finiteamplitude,
 author = {O. Dauchot and F. Daviaud},
 journal = {Phys. Fluids},
 pages = {335--343},
 title = {{F}inite-amplitude perturbation and spots growth mechanism in plane {C}ouette flow},
 volume = {7},
 year = {1995}
}

@incollection{dauchot1996pairs,
 author = {Dauchot, O. and Bottin, S. and Daviaud, F.},
 booktitle = {Advances in Turbulence VI},
 doi = {10.1007/978-94-009-0297-8\_92},
 editor = {Gavrilakis, S. and Machiels, L. and Monkewitz, P. A.},
 isbn = {978-94-010-6618-1},
 pages = {329--332},
 publisher = {Springer},
 title = {{P}airs of counter-rotating streamwise vortices: {A} finite amplitude solution in plane {C}ouette flow},
 volume = {36},
 year = {1996}
}

@article{dauchot1998experimental,
 author = {O. Dauchot and F. Daviaud and P. Manneville},
 journal = {Phys. Fluids},
 pages = {2597--2607},
 title = {{E}xperimental evidence of streamwise vortices as finite amplitude solutions in transitional plane {C}ouette flow},
 volume = {10},
 year = {1998}
}

@article{dauchot2000phase,
 author = {O. Dauchot and N. Vioujard},
 doi = {10.1007/s100510050142},
 journal = {Eur. Phys. J. B},
 pages = {377--381},
 title = {{P}hase space analysis of a dynamical model for the subcritical transition to turbulence in plane {C}ouette flow},
 volume = {14},
 year = {2000}
}

@article{dauchot2012subcritical,
 author = {Dauchot, O. and Bertin, E.},
 bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevE.86.036312},
 doi = {10.1103/PhysRevE.86.036312},
 journal = {Phys. Rev. E},
 numpages = {10},
 pages = {036312},
 title = {{S}ubcritical transition to turbulence: {W}hat we can learn from the physics of glasses},
 volume = {86},
 year = {2012}
}

@article{davey1973on,
 author = {A. Davey},
 journal = {J. Fluid Mech.},
 pages = {369--380},
 title = {{O}n the stability of plane {C}ouette flow to infinitesimal disturbances},
 volume = {57},
 year = {1973}
}

@article{daviaud1992spatiotemporal,
 abstract = {Describe the occurrence of the spatio-temporal intermittency in a 1-d convective system that shows time-independent patterns based on the amplitude equation approach},
 author = {F. Daviaud and J. Lega and P. Berg{\'e} and P. Coullet and M. Dubois},
 journal = {Physica D},
 pages = {287--308},
 title = {{S}patio-temporal intermittency in a 1{D} convective pattern: {T}heoretical model and experiments},
 volume = {55},
 year = {1992}
}

@article{daviaud1992subcritical,
 author = {F. Daviaud and J. Hegseth and P. Berge},
 journal = {Phys. Rev. Lett.},
 pages = {2511--2514},
 title = {{S}ubcritical transition to turbulence in plane {C}ouette flow},
 volume = {69},
 year = {1992}
}

@article{davidchack1999efficient,
 author = {R. L. Davidchack and Y.-C. Lai},
 journal = {Phys. Rev. E},
 pages = {6172},
 title = {{E}fficient algorithm for detecting unstable periodic orbits in chaotic systems},
 volume = {60},
 year = {1999}
}

@article{davis2011optimal,
 abstract = {This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun--Earth, Earth--Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost is accompanied by an increase in transfer time of flight.},
 author = {Davis, K. E. and Anderson, R. L. and Scheeres, D. J. and Born, G. H.},
 doi = {10.1007/s10569-010-9327-x},
 journal = {Celestial Mech. Dynam. Astronom.},
 pages = {241--264},
 title = {{O}ptimal transfers between unstable periodic orbits using invariant manifolds},
 volume = {109},
 year = {2011}
}

@book{dayan2001theoretical,
 address = {Cambridge, MA},
 author = {Dayan, P. and Abbott, L. F.},
 publisher = {MIT Press},
 title = {{T}heoretical {N}euroscience},
 year = {2001}
}

@article{deardorff1963on,
 abstract = {The problem of the stability of plane Couette
flow to infinitesimal disturbances is carried numerically
to larger Reynolds numbers than heretofore. The flow is
definitely stable up to R = 1430},
 author = {J. W. Deardorff},
 journal = {J. Fluid Mech.},
 pages = {623--630},
 title = {{O}n the stability of viscous plane {C}ouette flow},
 volume = {15},
 year = {1963}
}

@article{debroglie1923ondes,
 author = {L. de Broglie},
 journal = {C. R.},
 publisher = {Acad. Sci. Paris},
 title = {{O}ndes et quanta},
 volume = {177},
 year = {1923}
}

@article{degn1979bistability,
 author = {H. Degn and L. F. Olsen and J. W. Perram},
 doi = {10.1111/j.1749-6632.1979.tb29503.x},
 journal = {Ann. N. Y. Acad. Sci.},
 pages = {623--637},
 title = {{B}istability, oscillation, and chaos in an enzyme reaction},
 volume = {316},
 year = {1979}
}

@incollection{deguchi2012exact,
 addr = {New York},
 author = {Deguchi, K. and Nagata, M.},
 bdsk-url-1 = {http://dx.doi.org/10.1007/978-3-642-28968-2\_27},
 booktitle = {Progress in Turbulence and Wind Energy IV},
 doi = {10.1007/978-3-642-28968-2\_27},
 editor = {Oberlack, M. and Peinke, J. and Talamelli, A. and Castillo, L. and H{\"o}lling, M.},
 pages = {129--134},
 publisher = {Springer},
 series = {Proceedings in Physics},
 title = {{E}xact coherent state with travelling hairpin vortex in plane {C}ouette flow},
 volume = {141},
 year = {2012}
}

@article{deguchi2013emergence,
 author = {K. Deguchi and P. Hall and A. Walton},
 doi = {10.1017/jfm.2013.27},
 journal = {J. Fluid Mech.},
 pages = {58--85},
 title = {{T}he emergence of localized vortex-wave interaction states in plane {C}ouette flow},
 volume = {721},
 year = {2013}
}

@article{deguchi2014canonical,
 author = {Deguchi, K. and Hall, P.},
 doi = {10.1098/rsta.2013.0352},
 journal = {Phil. Trans. Royal Soc. A},
 number = {2020},
 pages = {20130352},
 title = {{C}anonical exact coherent structures embedded in high {R}eynolds{S} number flows},
 volume = {372},
 year = {2014}
}

@article{deguchi2014subcritical,
 author = {Deguchi, K. and Meseguer, A. and Mellibovsky, F.},
 doi = {10.1103/PhysRevLett.112.184502},
 journal = {Phys. Rev. Lett.},
 pages = {184502},
 title = {{S}ubcritical equilibria in {T}aylor-{C}ouette flow},
 volume = {112},
 year = {2014}
}

@article{deguchi2015selfsustained,
 author = {Deguchi, K.},
 doi = {10.1017/jfm.2015.514},
 journal = {J. Fluid Mech.},
 pages = {R6},
 title = {{S}elf-sustained states at {K}olmogorov microscale},
 volume = {781},
 year = {2015}
}

@article{delozar2012edge,
 author = {de Lozar, A. and Mellibovsky, F. and Avila, M. and Hof, B.},
 doi = {10.1103/PhysRevLett.108.214502},
 journal = {Phys. Rev. Lett.},
 pages = {214502},
 title = {{E}dge state in pipe flow experiments},
 volume = {108},
 year = {2012}
}

@article{dembo1982inexact,
 author = {R. S. Dembo and S. C. Eisenstat and T. Steihaug},
 journal = {SIAM J. Numer. Anal.},
 pages = {400--408},
 title = {{I}nexact {N}ewton methods},
 volume = {19},
 year = {1982}
}

@article{demmel2000computing,
 author = {J.W. Demmel and L. Dieci and M.J. Friedman},
 journal = {SIAM J. Sci. Comput.},
 pages = {81--94},
 title = {{C}omputing connecting orbits via an improved algorithm for continuing invariant spaces},
 volume = {22},
 year = {2000}
}

@book{dennis1996numerical,
 addr = {Philadelphia},
 author = {J.E. Dennis and R.B. Schnabel},
 publisher = {SIAM},
 title = {{N}umerical {M}ethods for {U}nconstrained {O}ptimization and {N}onlinear {E}quations},
 year = {1996}
}

@article{dennis2014distinct,
 author = {Dennis, D. J. C. and Sogaro, F. M.},
 doi = {10.1103/PhysRevLett.113.234501},
 journal = {Phys. Rev. Lett.},
 pages = {234501},
 title = {{D}istinct organizational states of fully developed turbulent pipe flow},
 volume = {113},
 year = {2014}
}

@techreport{departmentofenergy2001pump,
 author = {U.S. Department of Energy},
 title = {{P}ump {L}ife {C}ycle {C}osts: {A} Guide {T}o {L}CC {A}nalysis for {P}umping {S}ystems},
 url = {https://www.energy.gov/sites/prod/files/2014/05/f16/pumplcc\_1001.pdf},
 year = {2001}
}

@article{descalzi2013quasionedimensional,
 author = {Descalzi, O. and Brand, H. R.},
 bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevE.87.022915},
 doi = {10.1103/PhysRevE.87.022915},
 journal = {Phys. Rev. E},
 pages = {022915},
 title = {{Q}uasi-one-dimensional solutions and their interaction with two-dimensional dissipative solitons},
 volume = {87},
 year = {2013}
}

@article{dettmann1997cycle,
 author = {C. P. Dettmann and P. Cvitanovi{\' c}},
 journal = {Phys. Rev. E},
 pages = {6687},
 title = {{C}ycle expansions for intermittent diffusion},
 volume = {56},
 year = {1997}
}

@article{dettmann1997stability,
 author = {C. P. Dettmann and G. P. Morriss},
 bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevLett.78.4201},
 doi = {10.1103/PhysRevLett.78.4201},
 journal = {Phys. Rev. Lett.},
 pages = {4201--4204},
 title = {{S}tability ordering of cycle expansions},
 volume = {78},
 year = {1997}
}

@article{deuflhard1984computation,
 author = {P. Deuflhard},
 journal = {BIT},
 pages = {456},
 title = {{C}omputation of periodic solutions of nonlinear {O}DE},
 volume = {34},
 year = {1984}
}

@book{devaney1989introduction,
 address = {Reading, MA},
 author = {R. L. Devaney},
 publisher = {Addison-Wesley},
 title = {{A}n {I}ntroduction to {C}haotic {D}ynamical systems},
 year = {1989}
}

@book{devaney1992first,
 author = {R. L. Devaney},
 isbn = {9780813345475},
 publisher = {Avalon Publishing},
 series = {Studies in Nonlinearity},
 title = {{A} First {C}ourse {I}n {C}haotic {D}ynamical {S}ystems: {T}heory {A}nd {E}xperiment},
 year = {1992}
}

@article{dhont2013action,
 author = {G. Dhont and B. I. Zhilinski{\i}́},
 doi = {10.1088/1751-8113/46/45/455202},
 journal = {J. Phys. A},
 pages = {455202},
 title = {{T}he action of the orthogonal group on planar vectors: invariants, covariants and syzygies},
 volume = {46},
 year = {2013}
}

@article{dhont2015action,
 author = {G. Dhont and F. Patra and B. I. Zhilinski{\i}́s},
 doi = {10.1088/1751-8113/48/3/035201},
 journal = {J. Phys. A},
 pages = {035201},
 title = {{T}he action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of {M}olien functions},
 volume = {48},
 year = {2015}
}

@article{diaconis1984asymptotics,
 author = {Diaconis, P. and Freedman, D.},
 journal = {Ann. Statistics},
 pages = {793--815},
 title = {{A}symptotics of graphical projection pursuit},
 volume = {12},
 year = {1984}
}

@book{diacu1996celestial,
 address = {Princeton},
 author = {Diacu, F. and Holmes, P.},
 publisher = {Princeton Univ. Press},
 title = {{C}elestial {E}ncounters: {T}he {O}rigins of {C}haos and {S}tability},
 year = {1996}
}

@article{diakonos1998systematic,
 archiveprefix = {arXiv},
 author = {F. K. Diakonos and P. Schmelcher and O. Biham},
 eprint = {chao-dyn9810022},
 journal = {Phys. Rev. Lett.},
 title = {{S}ystematic computation of the least unstable periodic orbits in chaotic attractors},
 year = {1998}
}

@article{dieci2007error,
 author = {Dieci, L. and Jolly, M. S. and Rosa, R. and Van Vleck, E. S.},
 journal = {Discrete Contin. Dynam. Systems},
 pages = {555--580},
 title = {{E}rror in approximation of {L}yapunov exponents on inertial manifolds: {T}he {K}uramoto-{S}ivashinsky equation},
 volume = {9},
 year = {2007}
}

@unpublished{dijkstra2013numerical,
 author = {Dijkstra, H. A. and Wubs, F. W. and Cliffe, A. K. and Doedel, E. and Dragomirescu, I. F. and Eckhardt, B. and Gelfgat, A. Y. and Hazel, A. L. and Lucarini, V. and Salinger, A. G. and Phipps, E. T. and Sanchez-Umbria, J. and Schuttelaars, H. and Tuckerman, L. S. and Thiele, U.},
 note = {Commun. Comput. Phys., to appear},
 title = {{N}umerical bifurcation methods and their application to fluid dynamics: {A}nalysis beyond simulation},
 year = {2013}
}

@article{ding2016estimating,
 author = {Ding, X. and Chat{\'e}, H. and Cvitanovi\textbackslash          ,́ P. and Siminos, E. and Takeuchi, K. A.},
 doi = {10.1103/PhysRevLett.117.024101},
 journal = {Phys. Rev. Lett.},
 pages = {024101},
 title = {{E}stimating the dimension of the inertial manifold from unstable periodic orbits},
 volume = {117},
 year = {2016}
}

@misc{ding2016integration,
 author = {X. Ding and S. H. Kang},
 note = {In preparation},
 title = {{I}ntegration of a cubic-quintic complex {G}inzburg--{L}andau exploding soliton},
 year = {2016}
}

@article{ding2016periodic,
 author = {Ding, X. and Cvitanovi{\' c}, P.},
 doi = {10.1137/15M1037299},
 journal = {SIAM J. Appl. Dyn. Syst.},
 pages = {1434--1454},
 title = {{P}eriodic eigendecomposition and its application in {K}uramoto-{S}ivashinsky system},
 volume = {15},
 year = {2016}
}

@misc{ding2016relative,
 author = {Ding, X. and Cvitanovi{\' c}, P.},
 note = {In preparation},
 title = {{R}elative periodic orbit explosion in cubic-quintic complex {G}inzburg--{L}andau equation},
 year = {2016}
}

@book{dittrich2001classical,
 address = {New York},
 author = {W. Dittrich and M. Reuter},
 publisher = {Springer},
 title = {{C}lassical and {Q}uantum {D}ynamics: {F}rom {C}lassical {P}aths to {P}ath {I}ntegrals},
 year = {2001}
}

@article{doebeli2014chaos,
 archiveprefix = {arXiv},
 author = {M. Doebeli and I. Ispolatov},
 doi = {10.1111/evo.12354},
 eprint = {1309.6261},
 journal = {Evolution},
 number = {5},
 pages = {1365-1373},
 title = {{C}haos and {U}npredictability in {E}volution},
 volume = {68},
 year = {2014}
}

@article{doebner1994properties,
 author = {H.-D. Doebner and G. A. Goldin},
 journal = {J. Phys. A},
 pages = {1771--1780},
 title = {{P}roperties of nonlinear {S}chr{\"o}dinger equations associated with diffeomorphism group representations},
 volume = {27},
 year = {1994}
}

@article{doedel1991numerical,
 author = {E. Doedel and H. B. Keller and J. P. Kernevez},
 doi = {10.1142/S0218127491000397},
 journal = {Int. J. Bifurcat. Chaos},
 number = {03},
 pages = {493-520},
 title = {{N}umerical {A}nalysis and {C}ontrol of {B}ifurcation {P}roblems ({I}): {B}ifurcation in {F}inite {D}imensions},
 volume = {01},
 year = {1991}
}

@article{doelman1991finitedimensional,
 author = {A. Doelman},
 journal = {Nonlinearity},
 pages = {231--250},
 title = {{F}inite-dimensional models of the {G}inzburg-{L}andau equation},
 volume = {4},
 year = {1991}
}

@article{doelman1996breaking,
 author = {A. Doelman},
 journal = {Physica D},
 pages = {398--428},
 title = {{B}reaking the hidden symmetry in the {G}inzburg-{L}andau equation},
 volume = {97},
 year = {1996}
}

@article{doering1987exact,
 author = {C. R. Doering and J. D. Gibbon and D. D. Holm and B. Nikolaenko},
 journal = {Phys. Rev. Lett.},
 pages = {2911--2914},
 title = {{E}xact {L}yapunov dimension of the universal attractor for the {C}omplex {G}inzburg-{L}andau equation},
 volume = {59},
 year = {1987}
}

@article{doering1988lowdimensional,
 abstract = {The stability of plane waves of the CGLe is analyzed in a
periodic domain. The mass and energy integrals are estimated to give
an upper bound of the magnitude of the field variable. After that,
cone condition is established for the CGLe which indicates the
existence of the inertial manifold and proves the existence of finite
dimensionality of the attractor in this infinite-dimensional system.
The Lyapunov dimension is estimated. Note the defition of Lyapunov
exponents which is a little different from usual.},
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 year = {1982}
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@phdthesis{gibson2002dynamical,
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@article{gibson2008visualizing,
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@article{gibson2009equilibrium,
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@unpublished{gibson2010periodic,
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@techreport{gibson2013channelflow,
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 title = {{C}hannelflow: {A} spectral {N}avier-{S}tokes simulator in {C}++},
 year = {2013}
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@misc{gibson2013database,
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 year = {2013}
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@article{gibson2014spanwiselocalized,
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@techreport{gibson2015movies,
 author = {J. F. Gibson and P. Cvitanovi{\' c}},
 institution = {Georgia Inst. of Technology},
 publisher = {Center for Nonlinear Science},
 title = {{M}ovies of plane {C}ouette},
 url = {http://ChaosBook.org/tutorials},
 year = {2015}
}

@misc{gibson2020channelflow,
 author = {J. F. Gibson and F. Reetz and S. Azimi and
A. Ferraro and T. Kreilos and H. Schrobsdorff and
M. Farano and A. F. Yesil and S. S. Sch{\"u}tz and
M. Culpo and T. M. Schneider},
 title = {{C}hannelflow 2.0 (in preparation)},
 year = {2020}
}

@book{gilmore2007symmetry,
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@unpublished{gilmore2010connecting,
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@unpublished{ginzburg2004existence,
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Statistical emphasis on the transport and mixing phenomena in fluids
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Two symbol sequence converges to the same cycle.},
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@book{guckenheimer1983nonlinear,
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@book{gutzwiller1990chaos,
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@phdthesis{halcrow2008geometry,
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 year = {2008}
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@book{hale1969ordinary,
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@book{hale1984introduction,
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@book{hale1991dynamics,
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@book{hall2003lie,
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@article{haller1998reduction,
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@article{haller2015lagrangian,
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@book{halmos1958finitedimensional,
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 title = {{R}egeneration mechanisms of near-wall turbulence structures},
 volume = {287},
 year = {1995}
}

@article{hansen1992remarks,
 abstract = {Discuss the change of the symbol sequence of one
particular periodic orbit in the H{\'e}non map.},
 author = {K. T. Hansen},
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 title = {{R}emarks on the symbolic dynamics for the {H}{\'e}non map},
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 year = {1992}
}

@phdthesis{hansen1993symbolic,
 address = {Box 1048, Blindern, N-0316, Oslo, Norway},
 author = {K. T. Hansen},
 note = {\textbackslash          tt ChaosBook.org/projects/KTHansen/thesis},
 school = {University of Oslo},
 title = {{S}ymbolic {D}ynamics in {C}haotic systems},
 year = {1993}
}

@article{hansen1995alternative,
 archiveprefix = {arXiv},
 author = {Hansen, K. T.},
 eprint = {chao-dyn9507003},
 journal = {Phys. Rev. E},
 title = {{A}lternative method to find orbits in chaotic systems},
 year = {1995}
}

@book{hao1990chaos,
 address = {Singapore},
 author = {B.-L. Hao},
 publisher = {World Scientific},
 title = {{C}haos {I}I},
 year = {1990}
}

@book{hao1998applied,
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 author = {B.-L. Hao and W.-M. Zheng},
 publisher = {World Scientific},
 title = {{A}pplied {S}ymbolic {D}ynamics and {C}haos},
 year = {1998}
}

@article{harris2013wavelike,
 author = {D. M. Harris and J. Moukhtar and E. Fort and Y. Couder and J. W. M. Bush},
 doi = {10.1103/PhysRevE.88.011001},
 issue = {1},
 journal = {Phys. Rev. E},
 pages = {011001},
 title = {{W}avelike statistics from pilot-wave dynamics in a circular corral},
 volume = {88},
 year = {2013}
}

@article{harris2014droplets,
 author = {D. M. Harris and J. W. M. Bush},
 doi = {10.1017/jfm.2013.627},
 journal = {J. Fluid Mech.},
 pages = {444\textendash          464},
 publisher = {Cambridge University Press},
 title = {{D}roplets walking in a rotating frame: from quantized orbits to multimodal statistics},
 volume = {739},
 year = {2014}
}

@article{harter1969algebraic,
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 journal = {J. Math. Phys.},
 pages = {739--752},
 title = {{A}lgebraic theory of ray representations of finite groups},
 volume = {10},
 year = {1969}
}

@article{harter1978doublegroup,
 author = {Harter, W. G. and Dos Santos, N.},
 journal = {Amer. J. Phys.},
 pages = {251--263},
 title = {{D}ouble-group theory on the half-shell and the two-level system. {I}. {R}otation and half-integral spin states},
 volume = {46},
 year = {1978}
}

@book{harter1993principles,
 address = {New York},
 author = {Harter, W. G.},
 publisher = {Wiley},
 title = {{P}rinciples of {S}ymmetry, {D}ynamics, and {S}pectroscopy},
 year = {1993}
}

@book{hartman1964ordinary,
 address = {New York},
 author = {P. Hartman},
 publisher = {Wiley-Interscience},
 title = {{O}rdinary {D}ifferential {E}quations},
 year = {1964}
}

@inproceedings{hegseth1992intermittent,
 address = {New York},
 author = {J. Hegseth and F. Daviaud and P. Berge},
 booktitle = {Ordered and Turbulent Patterns in Taylor-Couette Flow},
 editor = {C. D. Andereck and F. Hayot},
 pages = {159--166},
 publisher = {Plenum},
 title = {{I}ntermittent turbulence in plane and circular {C}ouette flow},
 year = {1992}
}

@article{heidel2007nonchaotic,
 author = {J. Heidel and F. Zhang},
 doi = {10.1142/S021812740701821X},
 eprint = {http://www.worldscientific.com/doi/pdf/10.1142/S021812740701821X},
 journal = {Int. J. Bifur. Chaos},
 pages = {2049--2072},
 title = {{N}onchaotic and chaotic behavior in three-dimensional quadratic systems: {F}ive-one conservative cases},
 volume = {17},
 year = {2007}
}

@article{heisenberg1924uber,
 author = {W. Heisenberg},
 journal = {Ann. Phys. Lpz.},
 note = {fix volume number},
 pages = {577},
 title = {{\"U}ber {S}tabilit{\"a}t und {T}urbulenz von {F}l{\"a}teitsstr{\"a}ser},
 volume = {74 44},
 year = {1924}
}

@article{heisenberg1948on,
 author = {Heisenberg, W.},
 journal = {Proc. R. Soc. London A},
 pages = {402--406},
 title = {{O}n the theory of statistical and isotropic turbulence},
 volume = {195},
 year = {1948}
}

@article{heisenberg1948zur,
 author = {Heisenberg, W.},
 journal = {Z. Phys.},
 note = {On the statistical theory of turbulence, translation by M. L. Mahler, National Advisory Committee for Aeronautics, Technical memorandum, no. 1431},
 pages = {628--657},
 title = {{Z}ur statistischen {T}heorie der {T}urbulenz},
 volume = {124},
 year = {1948}
}

@article{helleman1978periodic,
 author = {R. Helleman and T. Bountis},
 journal = {LNP},
 pages = {353},
 title = {{P}eriodic solutions of arbitrary period, variational methods},
 volume = {93},
 year = {1978}
}

@article{heller2008inflationary,
 archiveprefix = {arXiv},
 author = {E. J. Heller and L. Kaplan and F. Pollmann},
 eprint = {0712.4093},
 journal = {Proc. Natl. Acad. Sci. USA},
 pages = {7631},
 title = {{I}nflationary dynamics for matrix eigenvalue problems},
 volume = {105},
 year = {2008}
}

@article{heninger2015neighborhoods,
 archiveprefix = {arXiv},
 author = {J. M. Heninger and D. Lippolis and P. Cvitanovi{\' c}},
 doi = {10.1103/PhysRevE.92.062922},
 eprint = {1507.00462},
 journal = {Phys. Rev. E},
 pages = {062922},
 title = {{N}eighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system},
 volume = {92},
 year = {2015}
}

@misc{heninger2016perturbation,
 author = {J. M. Heninger and D. Lippolis and P. Cvitanovi{\' c}},
 note = {to be submitted to Comm. Nonlinear Sci. Numer. Simul.},
 title = {{P}erturbation theory for the {F}okker-{P}lanck operator in chaos},
 url = {http://arxiv.org/abs/1602.03044},
 year = {2016}
}

@article{henningson1994on,
 author = {D. Henningson and S. Reddy},
 journal = {Phys. Fluids},
 pages = {1396--1398},
 title = {{O}n the role of linear mechanisims in transition to turbulence},
 volume = {6},
 year = {1994}
}

@article{henon1964applicability,
 author = {M. H{\'e}non and C. Heiles},
 journal = {Astron. J.},
 pages = {73},
 title = {{T}he applicability of the third integral of motion: {S}ome numerical experiments},
 volume = {69},
 year = {1964}
}

@article{henon1976twodimensional,
 author = {M. H{\'e}non},
 doi = {10.1007/BF01608556},
 journal = {Commun. Math. Phys.},
 pages = {69--77},
 title = {{A} two-dimensional mapping with a strange attractor},
 volume = {50},
 year = {1976}
}

@book{henon2001generating,
 address = {New York},
 author = {M. H{\'e}non},
 publisher = {Springer},
 title = {{G}enerating {F}amilies in the {R}estricted {T}hree-{B}ody {P}roblem: {I}I. {Q}uantitative {S}tudy of {B}ifurcations},
 year = {2001}
}

@article{henry2000lattice,
 abstract = {A lattice refinement scheme is proposed based on the
linear approximation of a given map. Jacobian matrix is not
calculated directly but inferred from the neighboring points.},
 author = {B. I. Henry and S. D. Watt and S. L. Wearne},
 journal = {Anziam J.},
 pages = {C735--C751},
 title = {{A} lattice refinement scheme for finding periodic orbits},
 volume = {42E},
 year = {2000}
}

@article{herault2011periodic,
 author = {J. Herault and F. Rincon and C. Cossu and G. I. Ogilvie and P.-Y. Longaretti},
 journal = {Phys. Rev. E},
 pages = {036321},
 title = {{P}eriodic magnetorotational dynamo action as a prototype of nonlinear magnetic-field generation in shear flows},
 volume = {84},
 year = {2011}
}

@article{hethcote2000mathematics,
 author = {H. W. Hethcote},
 doi = {10.1137/S0036144500371907},
 journal = {SIAM Rev.},
 number = {4},
 pages = {599-653},
 title = {{T}he {M}athematics of {I}nfectious {D}iseases},
 volume = {42},
 year = {2000}
}

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 author = {Hilbert, D.},
 doi = {10.1007/BF01444162},
 journal = {Math. Ann.},
 pages = {313--373},
 title = {{\"U}ber die vollen {I}nvariantensysteme},
 volume = {42},
 year = {1893}
}

@book{hille1997ordinary,
 address = {New York},
 author = {E. Hille},
 publisher = {Dover},
 title = {{O}rdinary {D}ifferential {E}quations in the {C}omplex {D}omain},
 year = {1997}
}

@incollection{hindmarsh1983odepack,
 address = {Amsterdam},
 author = {Hindmarsh, A. C.},
 booktitle = {Scientific Computing},
 editor = {Stepleman, R. S.},
 pages = {55--64},
 publisher = {North-Holland},
 title = {O{D}EP{A}CK, a systematized collection of {O}DE solvers},
 url = {https://computation.llnl.gov/casc/nsde/pubs/u88007.pdf},
 volume = {1},
 year = {1983}
}

@article{hinrichsen2000nonequilibrium,
 author = {H. Hinrichsen},
 doi = {10.1080/00018730050198152},
 journal = {Adv Phys.},
 number = {7},
 pages = {815-958},
 publisher = {Taylor \& Francis},
 title = {{N}on-equilibrium critical phenomena and phase transitions into absorbing states},
 volume = {49},
 year = {2000}
}

@article{hiruta2017intermittent,
 archiveprefix = {arXiv},
 author = {Y. Hiruta and S. Toh},
 doi = {10.1103/PhysRevE.96.063112},
 eprint = {1706.04395},
 issue = {6},
 journal = {Phys. Rev. E},
 month = {Dec},
 numpages = {7},
 pages = {063112},
 publisher = {American Physical Society},
 title = {{I}ntermittent direction reversals of moving spatially localized turbulence observed in two-dimensional {K}olmogorov flow},
 url = {https://link.aps.org/doi/10.1103/PhysRevE.96.063112},
 volume = {96},
 year = {2017}
}

@article{hobson1993efficient,
 author = {Dana Hobson},
 doi = {10.1006/jcph.1993.1002},
 journal = {J. Comput. Phys.},
 number = {1},
 pages = {14--22},
 title = {{A}n {E}fficient {M}ethod for {C}omputing {I}nvariant {M}anifolds of {P}lanar {M}aps},
 url = {http://www.sciencedirect.com/science/article/pii/S0021999183710028},
 volume = {104},
 year = {1993}
}

@article{hof2003scaling,
 author = {B. Hof and A. Juel and T. Mullin},
 journal = {Phys. Rev. Lett.},
 pages = {244502},
 title = {{S}caling of the turbulence transition threshold in a pipe},
 volume = {91},
 year = {2003}
}

@article{hof2004experimental,
 abstract = {Transition to turbulence in pipe flow is one of the most
fundamental and longest-standing problems in fluid dynamics.
Stability theory suggests that the flow remains laminar for all flow
rates, but in practice pipe flow becomes turbulent even at moderate
speeds. This transition drastically affects the transport efficiency
of mass, momentum, and heat. On the basis of the recent discovery of
unstable traveling waves in computational studies of the
Navier-Stokes equations and ideas from dynamical systems theory, a
model for the transition process has been suggested. We report
experimental observation of these traveling waves in pipe flow,
confirming the proposed transition scenario and suggesting that the
dynamics associated with these unstable states may indeed capture the
nature of fluid turbulence.},
 author = {B. Hof and C. W. H. van Doorne and J. Westerweel and F. T. M. Nieuwstadt and H. Faisst and B. Eckhardt and H. Wedin and R. R. Kerswell and F. Waleffe},
 doi = {10.1126/science.1100393},
 journal = {Science},
 pages = {1594--1598},
 title = {{E}xperimental observation of nonlinear traveling waves in turbulent pipe flow},
 volume = {305},
 year = {2004}
}

@article{hof2005turbulence,
 author = {Hof, B. and van Doorne, C. W. and Westerweel, J. and Nieuwstadt, F. T.},
 journal = {Phys. Rev. Lett.},
 number = {21},
 pages = {214502},
 publisher = {APS},
 title = {{T}urbulence regeneration in pipe flow at moderate {R}eynolds numbers},
 volume = {95},
 year = {2005}
}

@article{hof2006finite,
 author = {Hof, B. and Westerweel, J. and Schneider, T. M. and Eckhardt, B.},
 journal = {Nature},
 pages = {59--62},
 title = {{F}inite lifetime of turbulence in shear flows},
 volume = {443},
 year = {2006}
}

@article{hof2008repeller,
 author = {Hof, B. and De Lozar, A. and Kuik, D.J. and Westerweel, J.},
 journal = {Phys. Rev. Lett.},
 pages = {214501},
 title = {{R}epeller or attractor? {S}electing the dynamical model for the onset of turbulence in pipe flow},
 volume = {101},
 year = {2008}
}

@article{hof2010eliminating,
 author = {B. Hof and A. de Lozar and M. Avila and X. Tu and T. M. Schneider},
 doi = {10.1126/science.1186091},
 journal = {Science},
 number = {5972},
 pages = {1491--1494},
 publisher = {American Association for the Advancement of Science},
 title = {{E}liminating {T}urbulence in {S}patially {I}ntermittent {F}lows},
 volume = {327},
 year = {2010}
}

@article{hof2017searching,
 author = {Hof, Bj{\"o}rn and Budanur, Nazmi Burak},
 journal = {Physics},
 pages = {25},
 publisher = {APS},
 title = {{S}earching for {O}rder in {T}urbulent {F}low},
 volume = {10},
 year = {2017}
}

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 author = {Hofbauer, F. and Keller, G.},
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 journal = {Math. Z.},
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 title = {{E}rgodic properties of invariant measures for piecewise monotonic transformations},
 volume = {180},
 year = {1982}
}

@book{hoffman1971linear,
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 author = {K. Hoffman and R. Kunze},
 edition = {Second},
 publisher = {Prentice-Hall},
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 year = {1971}
}

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 author = {P. Holmes and J. Marsden and J. Scheurle},
 journal = {Cont. Math., AMS},
 pages = {213--244},
 title = {{E}xponentially small splittings of separatrices with applications to {K}AM theory and degenerate bifurcations},
 volume = {81},
 year = {1988}
}

@book{holmes1996turbulence,
 address = {Cambridge},
 author = {P. Holmes and J. L. Lumley and G. Berkooz},
 publisher = {Cambridge Univ. Press},
 title = {{T}urbulence, {C}oherent {S}tructures, {D}ynamical {S}ystems and {S}ymmetry},
 year = {1996}
}

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 title = {{O}n the computation of invariant manifolds of fixed points},
 volume = {46},
 year = {1995}
}

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 author = {A. P. Hooper and R. Grimshaw},
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 pages = {37--45},
 title = {{N}onlinear instability at the interface between two viscous fluids},
 volume = {28},
 year = {1985}
}

@article{hooper1988travelling,
 author = {A. P. Hooper and R. Grimshaw},
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 journal = {Wave Motion},
 pages = {405--420},
 title = {{T}ravelling wave solutions of the {K}uramoto-{S}ivashinsky equation},
 volume = {10},
 year = {1988}
}

@unpublished{hopcroft2014foundations,
 author = {J. Hopcroft and R. Kannan},
 note = { REFhttp://\Hresearch.microsoft.com/en-US/people/kannan/book-no-solutions-aug-21-2014.pdf In preparation},
 title = {{F}oundations of {D}ata {S}cience},
 year = {2014}
}

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 author = {E. Hopf},
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 journal = {Commun. Pure Appl. Math.},
 pages = {303--322},
 title = {{A} mathematical example displaying features of turbulence},
 volume = {1},
 year = {1948}
}

@article{hopf1950partial,
 author = {E. Hopf},
 journal = {Commun. Pure Appl. Math.},
 pages = {201--230},
 title = {{T}he partial differential equation \textdollar          u\_t + uu\_x = u\_xx\textdollar          },
 volume = {3},
 year = {1950}
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 journal = {Sci. Am.},
 title = {{M}y {E}ncounter with the {L}ate {M}itchell {F}eigenbaum, {C}haos {P}ioneer and {C}ritic},
 year = {2019}
}

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 author = {K. Horiuti},
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 pages = {343--370},
 title = {{C}omparison of conservative and rotational forms in large eddy simulation of turbulent channel flow},
 volume = {71},
 year = {1987}
}

@book{hoyle2006pattern,
 address = {Cambridge},
 author = {R. Hoyle},
 publisher = {Cambridge Univ. Press},
 title = {{P}attern {F}ormation: {A}n {I}ntroduction to {M}ethods},
 year = {2006}
}

@article{huerre1990local,
 author = {Huerre, P. and Monkewitz, P. A.},
 doi = {10.1146/annurev.fl.22.010190.002353},
 journal = {Ann. Rev. Fluid Mech.},
 pages = {473--537},
 title = {{L}ocal and global instabilities in spatially developing flows},
 volume = {22},
 year = {1990}
}

@article{hughes1998improved,
 author = {Hughes, S. and Randriamampianina, A.},
 doi = {10.1002/(SICI)1097-0363(19980915)28:3<501::AID-FLD730>3.0.CO;2-S},
 journal = {Int. J. Numer. Meth. Fluids},
 pages = {501--521},
 title = {{A}n improved projection scheme applied to pseudospectral methods for the incompressible {N}avier-{S}tokes equations},
 volume = {28},
 year = {1998}
}

@article{humphreys2019fast,
 author = {D. P. Humphreys and M. R. McGuirl and M. Miyagi and A. J. Blumberg},
 doi = {10.1534/genetics.118.301565},
 eprint = {https://www.genetics.org/content/211/4/1191.full.pdf},
 issn = {0016-6731},
 journal = {Genetics},
 note = {\textbackslash          hrefhttps://www.biorxiv.org/content/10.1101/395210v1bioRxiv 395210},
 number = {4},
 pages = {1191--1204},
 publisher = {Genetics},
 title = {{F}ast {E}stimation of {R}ecombination {R}ates {U}sing {T}opological {D}ata {A}nalysis},
 url = {https://www.genetics.org/content/211/4/1191},
 volume = {211},
 year = {2019}
}

@article{hunt1996optimal,
 author = {B. R. Hunt and E. Ott},
 journal = {Phys. Rev. Lett.},
 pages = {2254},
 title = {{O}ptimal periodic orbits of chaotic systems},
 volume = {76},
 year = {1996}
}

@article{hurt1993three,
 author = {Hurt, N. E.},
 journal = {Rev. Math. Phys.},
 pages = {713--733},
 title = {{T}hree topics on periodic orbit theory: {A} review},
 volume = {5},
 year = {1993}
}

@article{hurt1993zeta,
 author = {Hurt, N. E.},
 doi = {10.1007/BF03323131},
 journal = {Results Math.},
 pages = {55--120},
 title = {{Z}eta functions and periodic orbit theory: {A} review},
 volume = {23},
 year = {1993}
}

@article{hussain1983coherent,
 author = {Hussain, A. K. M. F.},
 doi = {10.1063/1.864048},
 journal = {Phys. Fluids},
 pages = {2816--2850},
 title = {{C}oherent structures-reality and myth},
 year = {1983}
}

@article{hwang2016invariant,
 author = {Hwang, Y. and Willis, A. P. and Cossu, C.},
 doi = {10.1017/jfm.2016.470},
 journal = {J. Fluid Mech.},
 pages = {R1},
 title = {{I}nvariant solutions of minimal large-scale structures in turbulent channel flow for \textdollar          {R}e\_\tau          \textdollar           up to 1000},
 volume = {802},
 year = {2016}
}

@article{hyman1986kuramotosivashinsky,
 author = {J. M. Hyman and B. Nicolaenko},
 journal = {Physica D},
 pages = {113--126},
 title = {{T}he {K}uramoto-{S}ivashinsky equation: a bridge between {P}DE\textquotesingle          s and dynamical systems},
 volume = {18},
 year = {1986}
}

@article{hyman1986order,
 author = {J. M. Hyman and B. Nicolaenko and S. Zaleski},
 journal = {Physica D},
 pages = {265--292},
 title = {{O}rder and complexity in the {K}uramoto-{S}ivashinsky model of weakly turbulent interfaces},
 volume = {23},
 year = {1986}
}

@book{ince1956ordinary,
 address = {New York},
 author = {E. L. Ince},
 publisher = {Dover},
 title = {{O}rdinary {D}ifferential {E}quations},
 year = {1956}
}

@book{isham1999modern,
 address = {Singapore},
 author = {C. J. Isham},
 publisher = {World Scientific},
 title = {{M}odern {D}ifferential {G}eometry for {P}hysicists},
 year = {1999}
}

@article{ishimura2000nonexistence,
 author = {Ishimura, N. and Nakamura, M.},
 journal = {Taiwanese J. Math.},
 pages = {621--625},
 title = {{N}onexistence of monotonic solutions of some third-order {O}DE relevant to the {K}uramoto-{S}ivashinsky equation},
 url = {http://journal.tms.org.tw/index.php/TJM/article/view/1296},
 volume = {4},
 year = {2000}
}

@article{ishimura2002remarks,
 author = {N. Ishimura},
 doi = {10.1006/jdeq.2001.4018},
 issn = {0022-0396},
 journal = {J. Diff. Eqn.},
 pages = {466--477},
 title = {{R}emarks on third-order {O}DEs relevant to the {K}uramoto-{S}ivashinsky equation},
 volume = {178},
 year = {2002}
}

@article{itano2001dynamics,
 author = {T. Itano and S. Toh},
 journal = {J. Phys. Soc. Japan},
 pages = {701--714},
 title = {{T}he dynamics of bursting process in wall turbulence},
 volume = {70},
 year = {2001}
}

@article{itano2009hairpin,
 author = {Itano, T. and Generalis, S. C.},
 journal = {Phys. Rev. Lett.},
 pages = {114501},
 title = {{H}airpin vortex solution in planar {C}ouette flow: {A} tapestry of knotted vortices},
 volume = {102},
 year = {2009}
}

@book{ito1993encyclopedic,
 address = {Cambridge, MA},
 author = {It{\^o}, K.},
 publisher = {MIT Press},
 title = {{E}ncyclopedic {D}ictionary of {M}athematics},
 year = {1993}
}

@book{jackson1989perspectives,
 address = {Cambridge},
 author = {Jackson, E. Atlee},
 publisher = {Cambridge Univ. Press},
 title = {{P}erspectives of nonlinear dynamics: {V}ol. 1 and 2},
 year = {1989}
}

@article{jang1986on,
 author = {P. S. Jang and R. L. Gran and D. J. Benney},
 journal = {J. Fluid Mech.},
 pages = {109--123},
 title = {{O}n the origin of streamwise vortices in a turbulent boundary layer},
 volume = {169},
 year = {1986}
}

@article{janiaud1992eckhaus,
 author = {B. Janiaud and A. Pumir and D. Bensimon and V. Croquette and H. Richter and L. Kramer},
 journal = {Physica D},
 pages = {269--286},
 title = {{T}he {E}ckhaus instability for travelling waves},
 volume = {55},
 year = {1992}
}

@book{jeffreys1999methods,
 address = {Cambridge},
 author = {H. Jeffreys and B. Jeffreys},
 publisher = {Cambridge Univ. Press},
 title = {{M}ethods of {M}athematical {P}hysics},
 year = {1999}
}

@article{jensen1983complete,
 author = {Jensen, M. H. and Bak, P. and Bohr, T.},
 journal = {Phys. Rev. Lett.},
 pages = {1637--1639},
 title = {{C}omplete devil\textquotesingle          s staircase fractal dimension and universality of mode-locking structure in the circle map},
 volume = {50},
 year = {1983}
}

@article{jensen1984transition,
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 journal = {Phys. Rev. A},
 pages = {1960--1969},
 title = {{T}ransition to chaos by interaction of resonances in dissipative systems. {I}. {C}ircle maps},
 volume = {30},
 year = {1984}
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@article{jeong1997coherent,
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 journal = {J. Fluid Mech.},
 pages = {185--214},
 title = {{C}oherent structures near the wall in a turbulent boundary layer},
 volume = {332},
 year = {1997}
}

@article{jimenez1991minimal,
 author = {Jim{\'e}nez, J. and Moin, P.},
 doi = {10.1017/S0022112091002033},
 journal = {J. Fluid Mech.},
 pages = {213--240},
 title = {{T}he minimal flow unit in near-wall turbulence},
 volume = {225},
 year = {1991}
}

@article{jimenez1993structure,
 author = {J. Jim{\'e}nez and A. A. Wray and P.G. Saffman and R.S. Rogallo},
 doi = {10.1017/S0022112093002393},
 issn = {1469-7645},
 journal = {J. Fluid Mech.},
 numpages = {26},
 pages = {65--90},
 title = {{T}he structure of intense vorticity in isotropic turbulence},
 volume = {255},
 year = {1993}
}

@article{jimenez1999autonomous,
 author = {J. Jim{\'e}nez and A. Pinelli},
 doi = {10.1017/S0022112099005066},
 journal = {J. Fluid Mech},
 pages = {335--359},
 publisher = {Cambridge University Press},
 title = {{T}he autonomous cycle of near-wall turbulence},
 volume = {389},
 year = {1999}
}

@article{jimenez2001lowdimensional,
 author = {J. Jim{\'e}nez and M. P. Simens},
 doi = {10.1017/S0022112001004050},
 journal = {J. Fluid Mech},
 pages = {81--91},
 title = {{L}ow-dimensional dynamics of a turbulent wall flow},
 volume = {435},
 year = {2001}
}

@article{jimenez2005characterization,
 author = {J. Jim{\'e}nez and G. Kawahara and M. P. Simens and M. Nagata and M. Shiba},
 doi = {10.1063/1.1825451},
 journal = {Phys. Fluids},
 pages = {015105},
 title = {{C}haracterization of near-wall turbulence in terms of equilibrium and \textasciigrave          \textasciigrave          bursting\textquotesingle          \textquotesingle           solutions},
 volume = {17},
 year = {2005}
}

@article{jimenez2007what,
 author = {Jim{\'e}nez, J. and Moser, R. D.},
 doi = {10.1098/rsta.2006.1943},
 journal = {Phil. Trans. Royal Soc. A},
 pages = {715--732S},
 title = {{W}hat are we learning from simulating wall turbulence?},
 volume = {367},
 year = {2007}
}

@article{jimenez2011cascades,
 author = {J. Jim{\'e}nez},
 doi = {10.1146/annurev-fluid-120710-101039},
 journal = {Ann. Rev. Fluid Mech.},
 pages = {27-45},
 title = {{C}ascades in wall-bounded turbulence},
 volume = {44},
 year = {2011}
}

@article{jimenez2018coherent,
 archiveprefix = {arXiv},
 author = {J. Jim{\'e}nez},
 doi = {10.1017/jfm.2018.144},
 eprint = {1710.07493},
 journal = {J. Fluid Mech.},
 pages = {P1},
 publisher = {Cambridge University Press},
 title = {{C}oherent structures in wall-bounded turbulence},
 volume = {842},
 year = {2018}
}

@article{jimenez2018machineaided,
 archiveprefix = {arXiv},
 author = {J. Jim{\'e}nez},
 doi = {10.1017/jfm.2018.660},
 eprint = {1806.06247},
 journal = {J. Fluid Mech.},
 pages = {R1},
 publisher = {Cambridge University Press},
 title = {{M}achine-aided turbulence theory},
 volume = {854},
 year = {2018}
}

@article{johansson1991evolution,
 author = {Johansson, A. V. and Alfredsson, P. H. and Kim, J.},
 journal = {J. Fluid Mech.},
 pages = {579--599},
 publisher = {Cambridge University Press},
 title = {{E}volution and dynamics of shear-layer structures in near-wall turbulence},
 volume = {224},
 year = {1991}
}

@article{johnson2001oseberg,
 author = {M. E. Johnson and M. S. Jolly and I. G. Kevrekidis},
 journal = {Int. J. Bifurc. Chaos},
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 title = {{T}he {O}seberg transition: visualization of global bifurcations for the {K}uramoto-{S}ivashinsky equation},
 volume = {11},
 year = {2001}
}

@article{johnston2004accurate,
 author = {H. Johnston and J.-G. Liu},
 journal = {J. Comput. Physics},
 pages = {221--259},
 title = {{A}ccurate, stable and efficient {N}avier-{S}tokes solvers based on explicit treatment of the pressure term},
 volume = {199},
 year = {2004}
}

@book{jolliffe2002principal,
 author = {I. T. Jolliffe},
 doi = {10.1007/B98835},
 publisher = {Springer},
 title = {{P}rincipal {C}omponent {A}nalysis, {S}econd {E}dition},
 year = {2002}
}

@article{jones1987strong,
 author = {C. A. Jones and M. R. E. Proctor},
 bdsk-url-1 = {http://dx.doi.org/10.1016/0375-9601(87)90008-9},
 doi = {10.1016/0375-9601(87)90008-9},
 journal = {Phys. Lett. A},
 pages = {224--228},
 title = {{S}trong spatial resonance and travelling waves in {B}enard convection},
 volume = {121},
 year = {1987}
}

@article{jones1992steady,
 abstract = {For small c>0, at least one odd periodic solution
exists. For any c>0, there existes no monotone bounded
travelling wave solution. The normal form analysis may
not be able to give correct results to all orders.},
 author = {J. Jones and W. C. Troy and A. D. MacGillivary},
 doi = {10.1016/0022-0396(92)90143-B},
 journal = {J. Diff. Eqn.},
 pages = {28--55},
 title = {{S}teady solutions of the {K}uramoto-{S}ivashinsky equation for small wave speed},
 volume = {96},
 year = {1992}
}

@misc{jones2001scipy,
 author = {E. Jones and T. Oliphant and P. Peterson and others},
 title = {{S}ci{P}y: {O}pen source scientific tools for {P}ython},
 url = {http://www.scipy.org},
 year = {2001}
}

@article{joseph1963transverse,
 author = {D. D. Joseph and L. N. Tao},
 journal = {J. Appl. Mech.},
 pages = {147--148},
 title = {{T}ransverse velocity components in fully developed unsteady flows},
 volume = {30},
 year = {1963}
}

@article{jovanovic2014sparsitypromoting,
 archiveprefix = {arXiv},
 author = {Jovanovi{\' c}, M. R. and Schmid, P. J. and Nichols, J. W.},
 eprint = {1309.4165},
 journal = {Phys. Fluids},
 number = {2},
 pages = {024103},
 publisher = {American Institute of Physics},
 title = {{S}parsity-promoting dynamic mode decomposition},
 volume = {26},
 year = {2014}
}

@article{kadanoff1984escape,
 author = {L. Kadanoff and C. Tang},
 journal = {Proc. Natl. Acad. Sci. USA},
 pages = {1276},
 title = {{E}scape rate from strange repellers},
 volume = {81},
 year = {1984}
}

@article{kaiser2014clusterbased,
 archiveprefix = {arXiv},
 author = {E. Kaiser and B. R. Noack and L. Cordier and A. Spohn and M. Segond and M. Abel and G. Daviller and J. {\"O}sth and S. Krajnovi{\' c} and R. K. Niven},
 eprint = {1309.0524},
 journal = {J. Fluid Mech.},
 pages = {365--414},
 publisher = {Cambridge University Press},
 title = {{C}luster-based reduced-order modelling of a mixing layer},
 volume = {754},
 year = {2014}
}

@article{kaneda2003energy,
 author = {Y. Kaneda and T. Ishihara and M. Yokokawa and K. Itakura and A. Uno},
 doi = {http://dx.doi.org/10.1063/1.1539855},
 journal = {Phys. Fluids},
 number = {2},
 pages = {L21--L24},
 title = {{E}nergy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box},
 volume = {15},
 year = {2003}
}

@article{kaneda2006highresolution,
 author = { Y. Kaneda  and  T. Ishihara },
 doi = {10.1080/14685240500256099},
 journal = {J. Turbul.},
 number = {},
 pages = {N20},
 title = {{H}igh-resolution direct numerical simulation of turbulence},
 volume = {7},
 year = {2006}
}

@incollection{kaplan1979numerical,
 address = {New York},
 author = {Kaplan, J. L. and Yorke, J. A.},
 booktitle = {Functional Differential Equations and Approximation of Fixed Points},
 doi = {10.1007/BFb0064320},
 editor = {Peitgen, H.-O. and Walther, H.-O.},
 pages = {228--237},
 publisher = {Springer},
 title = {{N}umerical solution of a generalized eigenvalue problem for even mappings},
 volume = {730},
 year = {1979}
}

@article{kassam2005fourthorder,
 abstract = {A contour integral method is presented to evaluate
accurately the matrix functions with removable singularities.},
 author = {A.-K. Kassam and L. N. Trefethen},
 journal = {SIAM J. Sci. Comp.},
 pages = {1214--1233},
 title = {{F}ourth-order time stepping for stiff {P}DEs},
 volume = {26},
 year = {2005}
}

@article{kato2003unstable,
 abstract = {A UPO is found in the chaotic region of one shell turbulence model (GOY). The scaling exponents of the structure function are calculated to verify that the statistics are well approximated by the UPO.},
 author = {S. Kato and M. Yamada},
 journal = {Phys. Rev. E},
 pages = {025302},
 title = {{U}nstable periodic solutions embedded in a shell model turbulence},
 volume = {68},
 year = {2003}
}

@book{katok1995introduction,
 address = {Cambridge},
 author = {A. Katok and B. Hasselblatt},
 publisher = {Cambridge Univ. Press},
 title = {{I}ntroduction to the {M}odern {T}heory of {D}ynamical {S}ystems},
 year = {1995}
}

@article{kavousanakis2007projective,
 author = {M. E. Kavousanakis and R. Erban and A. G. Boudouvis and C. W. Gear and I. G. Kevrekidis},
 journal = {J. Comp. Phys.},
 keywords = {Coarse projective integration,Continuous symmetry,Dynamic renormalization,Multiscale computation,Projective integration},
 pages = {382--407},
 title = {{P}rojective and coarse projective integration for problems with continuous symmetries},
 volume = {225},
 year = {2007}
}

@article{kawahara2001periodic,
 abstract = {Two time-periodic solutions are found in a 3-d
constrained plane Couette flow, multishooting method being used.
The turbulent state mainly follows the periodic orbit with strong
variations. The gentle one is related to the bursting behavior of
the system. Heteroclinic orbits between these two periodic orbits
are found.},
 author = {G. Kawahara and S. Kida},
 doi = {10.1017/S0022112001006243},
 journal = {J. Fluid Mech.},
 pages = {291--300},
 title = {{P}eriodic motion embedded in plane {C}ouette turbulence: {R}egeneration cycle and burst},
 volume = {449},
 year = {2001}
}

@article{kawahara2005elementary,
 author = {G. Kawahara and S. Kida},
 journal = {Parity},
 pages = {12},
 title = {{E}lementary coherent structures in turbulent flows},
 volume = {20},
 year = {2005}
}

@article{kawahara2005laminarization,
 author = {G. Kawahara},
 journal = {Phys. Fluids},
 pages = {041702},
 title = {{L}aminarization of minimal plane {C}ouette flow: {G}oing beyond the basin of attraction of turbulence},
 volume = {17},
 year = {2005}
}

@inproceedings{kawahara2005unstable,
 address = {Dordrecht},
 author = {G. Kawahara and S. Kida and M. Nagata},
 booktitle = {One hundred years of boundary layer research},
 publisher = {Kluwer},
 title = {{U}nstable periodic motion in plane {C}ouette system: {T}he skeleton of turbulence},
 year = {2005}
}

@article{kawahara2006unstable,
 author = {G. Kawahara and S. Kida and L. van Veen},
 journal = {Nonlinear Process in Geophysics},
 pages = {499--507},
 title = {{U}nstable periodic motion in turbulent flows},
 volume = {13},
 year = {2006}
}

@article{kawahara2012significance,
 archiveprefix = {arXiv},
 author = {G. Kawahara and M. Uhlmann and L. van Veen},
 eprint = {1108.0975},
 journal = {Annu. Rev. Fluid Mech},
 pages = {203--225},
 title = {{T}he significance of simple invariant solutions in turbulent flows},
 volume = {44},
 year = {2012}
}

@article{kazantsev1983rydberg,
 author = {Kazantsev, A. P. and Pokrovsky, V. L. and Bergou, J.},
 bdsk-url-1 = {http://dx.doi.org/10.1103/PhysRevA.28.3659},
 doi = {10.1103/PhysRevA.28.3659},
 journal = {Phys. Rev. A},
 pages = {3659--3662},
 title = {{R}ydberg atoms in weak magnetic fields},
 volume = {28},
 year = {1983}
}

@article{kazantsev1998unstable,
 author = {Kazantsev, E.},
 bdsk-url-1 = {http://dx.doi.org/10.5194/npg-5-193-1998},
 doi = {10.5194/npg-5-193-1998},
 journal = {Nonlinear Processes in Geophysics},
 pages = {193--208},
 title = {{U}nstable periodic orbits and attractor of the barotropic ocean model},
 volume = {5},
 year = {1998}
}

@article{kazantsev2001sensitivity,
 author = {Kazantsev, E.},
 doi = {10.5194/npg-8-281-2001},
 journal = {Nonlinear Processes in Geophysics},
 pages = {281--300},
 title = {{S}ensitivity of the attractor of the barotropic ocean model to external influences: {A}pproach by unstable periodic orbits},
 volume = {8},
 year = {2001}
}

@article{kazantsev2001sensitivityb,
 abstract = {A description of a deterministic chaotic system in terms of unstable
periodic orbits (UPOs) is used to develop a method of an a priori
estimate of the sensitivity of statistical averages of the solution
to small external influences. This method allows us to determine
the forcing perturbation which maximizes the norm of the perturbation
of a statistical moment of the solution on the attractor. The method
was applied to the Lorenz model. The estimates of perturbations of
two statistical moments were compared with directly calculated values.
The comparison shows that some 100 UPOs are sufficient to realize
this approach and get a good accuracy. The linear approach remains
valid up to rather high norms of the forcing perturbation.},
 author = {Kazantsev, E.},
 bdsk-url-1 = {http://dx.doi.org/10.1016/S0960-0779(00)00154-5},
 doi = {10.1016/S0960-0779(00)00154-5},
 journal = {Chaos, Solitons \&\textbackslash           Fractals},
 pages = {1989--2005},
 title = {{S}ensitivity of attractor to external influences: {A}pproach by unstable periodic orbits},
 volume = {12},
 year = {2001}
}

@book{keeling2011modeling,
 author = {M. J. Keeling and P. Rohani},
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 publisher = {Princeton University Press},
 title = {{M}odeling {I}nfectious {D}iseases in {H}umans and {A}nimals},
 year = {2011}
}

@inproceedings{keller1977numerical,
 address = {New York},
 author = {H. B. Keller},
 booktitle = {Applications of Bifurcation Theory},
 editor = {P. H. Rabinowitz},
 pages = {359--384},
 publisher = {Academic Press},
 title = {{N}umerical solution of bifurcation and nonlinear eigenvalue problems},
 year = {1977}
}

@inproceedings{keller1979global,
 address = {New York},
 author = {H. B. Keller},
 booktitle = {Recent advances in numerical analysis},
 editor = {C. de Boor and G. H. Golub},
 pages = {73--94},
 publisher = {Academic Press},
 title = {{G}lobal homotopies and {N}ewton methods},
 year = {1979}
}

@book{keller1992numerical,
 address = {New York},
 author = {H. B. Keller},
 publisher = {Dover},
 title = {{N}umerical {M}ethods for {T}wo-{P}oint {B}oundary-{V}alue {P}roblems},
 year = {1992}
}

@book{kelley2003solving,
 address = {Philadelphia},
 author = {C. T. Kelley},
 isbn = {0898715466},
 pages = {118},
 publisher = {SIAM},
 title = {{S}olving {N}onlinear {E}quations with {N}ewton\textquotesingle          s {M}ethod},
 year = {2003}
}

@article{kent1992travellingwaves,
 abstract = {Some properties of the KSe are discussed. A unified
argument about the connections and periodic orbits is given. In
particular, the K-bifurcation of the system is conjectured to arises
in the 1:n resonances of a fixed point.},
 author = {P. Kent and J. Elgin},
 journal = {Nonlinearity},
 pages = {899--919},
 title = {{T}ravelling-waves of the {K}uramoto-{S}ivashinsky equation: period-multiplying bifurcations},
 volume = {5},
 year = {1992}
}

@inproceedings{kerber2016geometry,
 archiveprefix = {arXiv},
 author = {M. Kerber and D. Morozov and A. Nigmetov},
 booktitle = {Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX)},
 doi = {10.1137/1.9781611974317.9},
 eprint = {1606.03357},
 pages = {103-112},
 title = {{G}eometry {H}elps to {C}ompare {P}ersistence {D}iagrams},
 year = {2016}
}

@misc{kerber2016hera,
 author = {M. Kerber and D. Morozov and A. Nigmetov},
 note = {rlhttps://\ubitbucket.org/grey\_narn/hera},
 title = {{H}era},
 year = {2016}
}

@incollection{kerr2011everelusive,
 address = {New York},
 author = {Kerr, R. M. and Oliver, M.},
 booktitle = {An Invitation to Mathematics},
 doi = {10.1007/978-3-642-19533-4\_10},
 editor = {Schleicher, D. and Lackmann, M.},
 isbn = {978-3-642-19532-7},
 pages = {137--164},
 publisher = {Springer},
 title = {{T}he ever-elusive blowup in the mathematical description of fluids},
 year = {2011}
}

@article{kerswell2005recent,
 author = {R. R. Kerswell},
 doi = {10.1088/0951-7715/18/6/R01},
 journal = {Nonlinearity},
 pages = {R17--R44},
 title = {{R}ecent progress in understanding the transition to turbulence in a pipe},
 volume = {18},
 year = {2005}
}

@article{kerswell2007recurrence,
 abstract = { We find that travelling waves with low wall shear
stresses (lower branch solutions) are on a surface which separates initial
conditions which uneventfully relaminarise and those which lead to a
turbulent evolution. Evidence for recurrent travelling wave visits is
found in both 5D and 10D long periodic pipes but only for those travelling
waves with low-to-intermediate wall shear stress and for less than about
10\% of the time in turbulent flow. Dynamical structures such as periodic
orbits need to be isolated and included in any such expansion.},
 archiveprefix = {arXiv},
 author = {Kerswell, R. R. and Tutty, O.R.},
 doi = {10.1017/S0022112007006301},
 eprint = {physics/0611009},
 journal = {J. Fluid Mech.},
 pages = {69--102},
 title = {{R}ecurrence of travelling waves in transitional pipe flow},
 volume = {584},
 year = {2007}
}

@article{kerswell2014optimization,
 author = {R. R. Kerswell and C. C. T. Pringle and A. P. Willis},
 journal = {Rep. Prog. Phys.},
 number = {8},
 pages = {085901},
 title = {{A}n optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar},
 url = {http://stacks.iop.org/0034-4885/77/i=8/a=085901},
 volume = {77},
 year = {2014}
}

@article{kevrekidis1990back,
 author = {I. G. Kevrekidis and B. Nicolaenko and J. C. Scovel},
 journal = {SIAM J. Appl. Math.},
 pages = {760--790},
 title = {{B}ack in the saddle again: a computer assisted study of the {K}uramoto-{S}ivashinsky equation},
 volume = {50},
 year = {1990}
}

@article{khapko2014complexity,
 author = {Khapko, T. and Duguet, Y. and Kreilos, T. and Schlatter, P. and Eckhardt, B. and Henningson, D. S.},
 doi = {10.1140/epje/i2014-14032-3},
 journal = {Eur. Phys. J. E},
 pages = {32},
 title = {{C}omplexity of localised coherent structures in a boundary-layer flow},
 volume = {37},
 year = {2014}
}

@article{khapko2016edge,
 author = {T. Khapko and T. Kreilos and P. Schlatter and Y. Duguet and B. Eckhardt and D. S. Henningson},
 doi = {10.1017/jfm.2016.434},
 journal = {J. Fluid Mech.},
 publisher = {Cambridge University Press},
 title = {{E}dge states as mediators of bypass transition in boundary-layer flows},
 volume = {801},
 year = {2016}
}

@article{kim1971production,
 author = {Kim, H.T. and Kline, S.J. and Reynolds, W.C.},
 journal = {J. Fluid Mech.},
 pages = {133--160},
 title = {{T}he production of turbulence near a smooth wall in a turbulent boundary layer},
 volume = {50},
 year = {1971}
}

@article{kim1987turbulence,
 author = {J. Kim and P. Moin and R. Moser},
 doi = {10.1017/S0022112087000892},
 journal = {J. Fluid Mech.},
 pages = {133--166},
 title = {{T}urbulence statistics in fully developed channel flow at low {R}eynolds number},
 volume = {177},
 year = {1987}
}

@article{kim1999very,
 author = {K. C. Kim and R. J. Adrian},
 journal = {Phys. Fluids},
 pages = {417--422},
 title = {{V}ery large-scale motion in the outer layer},
 volume = {11},
 year = {1999}
}

@article{kim2003control,
 author = {J. Kim},
 journal = {Phys. Fluids},
 pages = {1093--1105},
 title = {{C}ontrol of turbulent boundary layers},
 volume = {15},
 year = {2003}
}

@article{kim2010development,
 author = {J. H. Kim and M. Nishihara and I. V. Adamovich and M. Samimy and S. V. Gorbatov and F. V. Pliavaka},
 doi = {10.1007/s00348-010-0819-y},
 journal = {Exp. Fluids},
 pages = {497--511},
 title = {{D}evelopment of localized arc-filament {R}F plasma actuators for high-speed and high {R}eynolds number flow control},
 volume = {49},
 year = {2010}
}

@article{kirby1992reconstructing,
 author = {M. Kirby and D. Armbruster},
 journal = {Z. Angew. Math. Phys.},
 pages = {999--1022},
 title = {{R}econstructing phase space from {P}DE simulations},
 volume = {43},
 year = {1992}
}

@article{kleinman2010shearflow,
 author = {R. R. Kleinman and D. J. Bodony and J. B. Freund},
 journal = {Phys. Fluids},
 pages = {116103},
 title = {{S}hear-flow excitation mechanisms of recessed localized arc-filament plasma actuators},
 volume = {22},
 year = {2010}
}

@inproceedings{kleiser1980treatment,
 address = {Viewweg, Braunschweig},
 author = {L. Kleiser and U. Schumann},
 booktitle = {Proc. 3\textbackslash          textsuperscriptrd GAMM Conf. Numerical Methods in Fluid Mechanics},
 editor = {E. Hirschel},
 organization = {GAMM},
 pages = {165--173},
 title = {{T}reatment of incompressibility and boundary conditions in 3-{D} numerical spectral simulations of plane channel flows},
 year = {1980}
}

@article{kleiser1991numerical,
 author = {L. Kleiser and T. A. Zang},
 journal = {Annu. Rev. Fluid Mech.},
 pages = {495--537},
 title = {{N}umerical simulation of transition in wall-bounded shear flows},
 volume = {23},
 year = {1991}
}

@article{klewicki1995viscous,
 author = {Klewicki, J. C. and Metzger, M. M. and Kelner, E. and E. M. Thurlow},
 journal = {Phys. Fluids},
 pages = {857},
 title = {{V}iscous sublayer flow visualizations at \textdollar          {R}\_{\th}eta \sim           1500000\textdollar          },
 volume = {7},
 year = {1995}
}

@article{kline1967structure,
 author = {Kline, S. J. and Reynolds, W. C. and Schraub, F. A. and Runstadler, P. W.},
 journal = {J. Fluid Mech.},
 pages = {741--773},
 title = {{T}he structure of turbulent boundary layers},
 volume = {30},
 year = {1967}
}

@article{knobloch2015spatial,
 author = {Knobloch, E.},
 doi = {10.1146/annurev-conmatphys-031214-014514},
 journal = {Ann. Math.. Rev. Condens. Matter Phys.},
 pages = {325--359},
 title = {{S}patial localization in dissipative systems},
 volume = {6},
 year = {2015}
}

@article{koch1986direct,
 author = {W. Koch},
 journal = {Acta Mechanica},
 pages = {11},
 title = {{D}irect resonances in {O}rr-{S}ommerfeld problems},
 volume = {59},
 year = {1986}
}

@article{kolmogorov1991local,
 abstract = {Study the distribution of velocity difference of
neighboring points based on the locally homogeneous and locally
isotropic hypothesized poperties of turbulence motion. Two more
similarity hypothesis suggested the existence of a universal
distribution for different viscosity and energy dissipation rate.
The large distance asymptotic behavior of the distribution is also
deduced.},
 author = {A. N. Kolmogorov},
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 abstract = {An algorithm of Newton\textquotesingle          s method is presented to find
periodic orbits for Lagrangian mappings. The method is based on
block-diagonalization of the Hessian Matrix of the action function.},
 author = {H-T Kook and J. D. Meiss},
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 pages = {317--326},
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 year = {1989}
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 abstract = {By constructing a 2N-dimensional symplectic map
for a Langrangian system, the orbit structure of the phase space
is discussed for a time-reversible system. Periodic orbits are
classfied according to the rotation number, symmetry, and morse index.
Unstable orbits repel other periodic orbits to form resonances
where a orbit can be trapped for long with the same rotation number.
The chain of resonances with the same commensurability forms channel
which provides the communication bridge between different resonances.
The connection of the orbit structure with the continued fraction
representation of real numbers is discussed.},
 author = {H-T. Kook and J. D. Meiss},
 journal = {Physica D},
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 title = {{P}eriodic orbits for reversible symplectic mappings},
 volume = {35},
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@misc{kornyak2007symmetries,
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 author = {Vladimir V. Kornyak},
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 title = {{S}ymmetries and dynamics of discrete systems},
 year = {2007}
}

@article{kramar2016analysis,
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 author = {M. Kram{\'a}r and R. Levanger and J. Tithof and B. Suri and M. Xu and M. Paul and M. F. Schatz and K. Mischaikow},
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 journal = {Physica D},
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 title = {{A}nalysis of {K}olmogorov flow and {R}ayleigh\textendash          {B}{\'e}nard convection using persistent homology},
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 year = {2016}
}

@article{krasikov2004growing,
 abstract = {The random sequence has been generalized both in the form of addition and in the probability distributions. Critical parameter values are discovered around which the system has very different behavior.},
 author = {I. Krasikov and G. J. Rodgers and C. E. Tripp},
 journal = {J. Phys. A},
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 title = {{G}rowing random sequences},
 volume = {37},
 year = {2004}
}

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 address = {New York},
 author = {B. Krauskopf and H. M. Osinga},
 booktitle = {Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems},
 editor = {E. Doedel and L. Tuckerman},
 pages = {199--208},
 publisher = {Springer},
 series = {The IMA Volumes in Mathematics and its Applications},
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 year = {2000}
}

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 doi = {10.1142/S0218127405012533},
 journal = {Int. J. Bifurc. Chaos},
 number = {03},
 pages = {763-791},
 title = {{A} Survey {O}f {M}ethods for {C}omputing ({U}n)stable {M}anifolds of {V}ector {F}ields},
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 year = {2005}
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@incollection{krauss2013probability,
 address = {New York},
 author = {Krauss, J. and Ertun{\c c}, {\"O}. and Ostwald, C. and Lienhart, H. and Delgado, A.},
 journal = {New Results in Numerical and Experimental Fluid Mechanics VIII },
 pages = {271--278},
 publisher = {Springer},
 title = {{P}robability of occurrence of the turbulent flow structures in transitional pipe flows},
 year = {2013}
}

@article{kreilos2012periodic,
 archiveprefix = {arXiv},
 author = {T. Kreilos and B. Eckhardt},
 doi = {10.1063/1.4757227},
 eprint = {1205.0347},
 journal = {Chaos},
 pages = {047505},
 title = {{P}eriodic orbits near onset of chaos in plane {C}ouette flow},
 volume = {22},
 year = {2012}
}

@article{kreilos2013edge,
 archiveprefix = {arXiv},
 author = {Kreilos, T. and Veble, G. and Schneider, T. M. and Eckhardt, B.},
 bdsk-url-1 = {http://dx.doi.org/10.1017/jfm.2013.212},
 doi = {10.1017/jfm.2013.212},
 eprint = {1209.0593},
 journal = {J. Fluid M.},
 pages = {100--122},
 title = {{E}dge states for the turbulence transition in the asymptotic suction boundary layer},
 volume = {726},
 year = {2013}
}

@article{kreilos2014comoving,
 archiveprefix = {arXiv},
 author = {T. Kreilos and S. Zammert and B. Eckhardt},
 doi = {10.1017/jfm.2014.305},
 eprint = {1309.4590},
 issn = {1469-7645},
 journal = {J. Fluid Mech.},
 pages = {685--697},
 title = {{C}omoving frames and symmetry-related motions in parallel shear flows},
 volume = {751},
 year = {2014}
}

@article{kreilos2014increasing,
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 issue = {4},
 journal = {Phys. Rev. Lett.},
 month = {Jan},
 numpages = {5},
 pages = {044503},
 publisher = {American Physical Society},
 title = {{I}ncreasing {L}ifetimes and the {G}rowing {S}addles of {S}hear {F}low {T}urbulence},
 url = {https://link.aps.org/doi/10.1103/PhysRevLett.112.044503},
 volume = {112},
 year = {2014}
}

@phdthesis{kreilos2014turbulence,
 author = {Kreilos, T.},
 note = {http://archiv.ub.uni-marburg.de/diss/z2014/0356/pdf/dtk.pdf},
 school = {Philipps-Universit{\"a}t Marburg},
 title = {{T}urbulence {T}ransition in {S}hear {F}lows and {D}ynamical {S}ystems {T}heory},
 year = {2014}
}

@article{kreilos2016localized,
 author = {T. Kreilos and J. F. Gibson and T. M. Schneider},
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 journal = {J. Fluid Mech.},
 pages = {R3},
 title = {{L}ocalized travelling waves in the asymptotic suction boundary layer},
 volume = {795},
 year = {2016}
}

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 year = {2018}
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@article{kuramoto1978diffusioninduced,
 abstract = {Phase turbulence and amplitude turbulence
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@book{kuramoto1984chemical,
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@article{kurianski2017simulations,
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 journal = {Phys. Rev. Fluids},
 month = {Nov},
 numpages = {14},
 pages = {113602},
 publisher = {American Physical Society},
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@book{kutz2016dynamic,
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 volume = {149},
 year = {2016}
}

@book{kuznetsov2010elements,
 author = {Y. Kuznetsov},
 isbn = {9781441919519},
 lccn = {2004046862},
 publisher = {Springer New York},
 series = {Applied Mathematical Sciences},
 title = {{E}lements of {A}pplied {B}ifurcation {T}heory},
 url = {https://books.google.at/books?id=X39lcgAACAAJ},
 year = {2010}
}

@article{kuznetsova2004catastrophe,
 author = {A. Yu. Kuznetsova and A. P. Kuznetsov and C. Knudsen and E. Mosekilde},
 doi = {10.1142/S0218127404009995},
 journal = {Int. J. Bif. Chaos},
 pages = {1241--1266},
 title = {{C}atastrophe theoretic classification of nonlinear oscillators},
 volume = {14},
 year = {2004}
}

@phdthesis{labousse2014etude,
 author = {Matthieu Labousse},
 school = {Universit{\'e} Paris 6},
 title = {{E}tude d\textquotesingle          une dynamique {\`a} m{\'e}moire de chemin: une exp{\'e}rimentation th\textbackslash          {\'o}rique},
 year = {2014}
}

@article{labousse2016pilotwave,
 author = {M. Labousse and A. U. Oza and S. Perrard and J. W. M. Bush},
 doi = {10.1103/PhysRevE.93.033122},
 issue = {3},
 journal = {Phys. Rev. E},
 month = {Mar},
 numpages = {6},
 pages = {033122},
 publisher = {American Physical Society},
 title = {{P}ilot-wave dynamics in a harmonic potential: {Q}uantization and stability of circular orbits},
 url = {https://link.aps.org/doi/10.1103/PhysRevE.93.033122},
 volume = {93},
 year = {2016}
}

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 author = {Lagha, M.},
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 doi = {10.1063/1.2793143},
 journal = {Phys. Fluids},
 pages = {4108},
 title = {{M}odeling of plane {C}ouette flow. {I}I. {O}n the spreading of a turbulent spot},
 volume = {19},
 year = {2007}
}

@article{lagha2007modelingb,
 author = {M. Lagha and P. Manneville},
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 volume = {58},
 year = {2007}
}

@article{lagha2007modelingc,
 author = {Lagha, M. and Manneville, P.},
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 doi = {10.1063/1.2768946},
 journal = {Phys. Fluids},
 pages = {4105},
 title = {{M}odeling of plane {C}ouette flow. {I}. {L}arge scale flow around turbulent spots},
 volume = {19},
 year = {2007}
}

@book{lai2011transient,
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 author = {Y. Lai and T. T{\'e}l},
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 title = {{T}ransient {C}haos: {C}omplex {D}ynamics on {F}inite {T}ime {S}cales},
 year = {2011}
}

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@phdthesis{lan2004dynamical,
 address = {Atlanta},
 author = {Y. Lan},
 note = {\textbackslash          tt ChaosBook.org/projects/theses.html},
 school = {School of Physics, Georgia Inst. of Technology},
 title = {{D}ynamical systems approach to \textdollar          1-d\textdollar           spatiotemporal chaos -- {A} cyclist\textquotesingle          s view},
 year = {2004}
}

@article{lan2004stationary,
 author = {Y. Lan and N. Garnier and P. Cvitanovi{\' c}},
 journal = {Physica D},
 pages = {193--212},
 title = {{S}tationary modulated-amplitude waves in the 1{D} complex {G}inzburg-{L}andau equation},
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 year = {2004}
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@article{lan2004variational,
 archiveprefix = {arXiv},
 author = {Y. Lan and P. Cvitanovi{\' c}},
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 journal = {Phys. Rev. E},
 title = {{V}ariational method for finding periodic orbits in a general flow},
 year = {2004}
}

@article{lan2006variational,
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 author = {Y. Lan and C. Chandre and P. Cvitanovi{\' c}},
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 journal = {Phys. Rev. E},
 title = {{V}ariational method for locating invariant tori},
 year = {2006}
}

@article{lan2008unstable,
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 author = {Y. Lan and P. Cvitanovi{\' c}},
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 journal = {Phys. Rev. E},
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 title = {{U}nstable recurrent patterns in {K}uramoto-{S}ivashinsky dynamics},
 volume = {78},
 year = {2008}
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@book{lasota1994chaos,
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@book{lautrup2011physics,
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 year = {2011}
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@incollection{lauwerier1986twodimensional,
 address = {Princeton},
 author = {H. A. Lauwerier},
 booktitle = {Chaos},
 editor = {V. Holden},
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 publisher = {Princeton Univ. Press},
 title = {{T}wo-dimensional iterative maps},
 year = {1986}
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 year = {2001}
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 year = {2016}
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 year = {1995}
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@misc{levnajic2008ergodic,
 abstract = {We present a computational study of the
invariant sets visualization method based on ergodic
partition theory. The algorithms
for computation of the time averages of many L1 functions
are developed and employed producing the approximation of
the phase space ergodic partitioning. The method is
exposed in the context of discrete-time dynamical systems
producing a graphical representation of the phase space
in terms of the invariant set structure that gives a
substantial insight into the global and local properties
of the dynamics. We use the Chirikov standard map in
order to show the implementation of the method, followed
by applications to other multi-dimensional maps. We examine
the visualization of periodic sets using harmonic time
averages.},
 archiveprefix = {arXiv},
 author = {Levnaji{\' c}, Z. and Mezi\textbackslash          ,́ I.},
 eprint = {0805.4221},
 title = {{E}rgodic theory and visualization {I}: {V}isualization of ergodic partition and invariant sets},
 year = {2008}
}

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 year = {2006}
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@article{li2007polymer,
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 year = {2007}
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@article{li2009exact,
 author = {Li, Y. C. and Viswanath, D.},
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 journal = {Phys. Fluids},
 title = {{E}xact and asymptotic conditions on traveling wave solutions of the {N}avier-{S}tokes equations},
 volume = {21},
 year = {2009}
}

@article{li2009matrix,
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 author = {Li, Y. and Chevillard, L. and Eyink, G. and Meneveau, C.},
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 pages = {016305},
 title = {{M}atrix exponential-based closures forthe turbulent subgrid-scale stress tensor},
 volume = {79},
 year = {2009}
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@misc{libertnoyearsymplectic,
 archiveprefix = {arXiv},
 author = {Libert, A.-S. and Hubaux, C. and Carletti, T.},
 eprint = {arXiv:10055611},
 title = {{S}ymplectic integration of deviation vectors and chaos determination. {A}pplication to the {H}{\'e}non-{H}eiles model and to the restricted three-body problem}
}

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 journal = {Math. Methods Appl. Sci.},
 pages = {931--960},
 title = {{W}ell-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics},
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 year = {2007}
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 author = {Lim, C. and Montaldi, J. and Roberts, M.},
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 title = {{R}elative equilibria of point vortices on the sphere},
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@book{lin1955theory,
 address = {Cambridge},
 author = {Lin, C. C.},
 publisher = {Cambridge Univ. Press},
 title = {{T}he {T}heory of {H}ydrodynamic {S}tability},
 year = {1955}
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@article{linkmann2015sudden,
 author = {M. F. Linkmann and A. Morozov},
 doi = {10.1103/PhysRevLett.115.134502},
 issue = {13},
 journal = {Phys. Rev. Lett.},
 numpages = {5},
 pages = {134502},
 publisher = {American Physical Society},
 title = {{S}udden {R}elaminarization and {L}ifetimes in {F}orced {I}sotropic {T}urbulence},
 volume = {115},
 year = {2015}
}

@article{linot2020deep,
 author = {Linot, Alec J. and Graham, Michael D.},
 doi = {10.1103/PhysRevE.101.062209},
 issue = {6},
 journal = {Phys. Rev. E},
 month = {Jun},
 numpages = {8},
 pages = {062209},
 publisher = {American Physical Society},
 title = {{D}eep learning to discover and predict dynamics on an inertial manifold},
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 year = {2020}
}

@article{lippolis2010how,
 archiveprefix = {arXiv},
 author = {D. Lippolis and P. Cvitanovi{\' c}},
 doi = {10.1103/PhysRevLett.104.014101},
 eprint = {0902.4269},
 journal = {Phys. Rev. Lett.},
 pages = {014101},
 title = {{H}ow well can one resolve the state space of a chaotic map?},
 volume = {104},
 year = {2010}
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@misc{lippolisnoyearoptimal,
 author = {D. Lippolis and P. Cvitanovi{\' c}},
 note = {In preparation, 2011},
 title = {{O}ptimal resolution of the state space of a chaotic flow in presence of noise}
}

@misc{liu2005divorcing,
 author = {J.-G. Liu and J. Liu and R. L. Pego},
 title = {{D}ivorcing pressure from viscosity in incompressible {N}avier-{S}tokes dynamics},
 url = {https://arxiv.org/abs/math/0502549},
 year = {2005}
}

@incollection{llibre2015averaging,
 address = {Basel},
 author = {Llibre, J.},
 booktitle = {Central Configurations, Periodic Orbits, and Hamiltonian Systems},
 doi = {10.1007/978-3-0348-0933-7\_1},
 isbn = {978-3-0348-0932-0},
 language = {English},
 pages = {1--104},
 publisher = {Springer},
 series = {Advanced Courses in Mathematics - CRM Barcelona},
 title = {{T}he averaging theory for computing periodic orbits},
 year = {2015}
}

@article{lombardi2011growth,
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 doi = {10.1017/S0022112010005574},
 journal = {J. Fluid Mech.},
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 title = {{G}rowth and instability of a laminar plume in a strongly stratified environment},
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 journal = {J. Comput. Physics},
 pages = {308-326},
 title = {{A}n efficient spectral-projection method for the {N}avier-{S}tokes equations in cylindrical geometries {I}. {A}xisymmetric cases},
 volume = {139},
 year = {1998}
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@article{lopez2000determining,
 author = {Lopez, J. M. and Marques, F.},
 doi = {10.1007/PL00001507},
 journal = {Z. Angew. Math. Phys.},
 keywords = {Key words. Floquet theory, self-rotation number, parametric excitation, quasiperiodic flow, Taylor--Couette flow.},
 pages = {61--74},
 title = {{D}etermining the self-rotation number following a {N}eimark-{S}acker bifurcation in the periodically forced {T}aylor-{C}ouette flow},
 volume = {51},
 year = {2000}
}

@article{lopez2005finite,
 author = {J. M. Lopez and F. Marques},
 journal = {Physica D},
 pages = {168--191},
 title = {{F}inite aspect ratio {T}aylor-{C}ouette flow: {S}hil'nikov dynamics of 2-tori},
 volume = {211},
 year = {2005}
}

@article{lopez2006relative,
 abstract = {Define relative periodic orbits and use Fourier modes
to find them.},
 archiveprefix = {arXiv},
 author = {V. L{\'o}pez and P. Boyland and M. T. Heath and R. D. Moser},
 bdsk-url-1 = {http://dx.doi.org/10.1137/040618977},
 doi = {10.1137/040618977},
 eprint = {nlin/0408018},
 journal = {SIAM J. Appl. Dyn. Syst.},
 pages = {1042--1075},
 title = {{R}elative periodic solutions of the {C}omplex {G}inzburg-{L}andau equation},
 volume = {4},
 year = {2006}
}

@article{lord2012computing,
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 journal = {SIAM J. Sci. Comput.},
 pages = {B24--B43},
 title = {{C}omputing stochastic travelling waves},
 volume = {34},
 year = {2012}
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@article{lozanoduran2014timeresolved,
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 title = {{T}ime-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades},
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 year = {2014}
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@misc{ltextquotesinglevov2007universal,
 archiveprefix = {arXiv},
 author = {V. S. L\textquotesingle          vov and I. Procaccia and O. Rudenko},
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 title = {{U}niversal model of finite-{R}eynolds number turbulent flow in channels and pipes},
 year = {2007}
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@article{lucas2014spatiotemporal,
 archiveprefix = {arXiv},
 author = {Lucas, D. and Kerswell, R. R.},
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 doi = {10.1017/jfm.2014.270},
 eprint = {1308.3356},
 journal = {J. Fluid Mech.},
 pages = {518--554},
 title = {{S}patiotemporal dynamics in 2{D} Kolmogorov flow over large domains},
 volume = {750},
 year = {2014}
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@article{lucas2015recurrent,
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 eprint = {1406.1820},
 journal = {Phys. Fluids},
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 year = {2015}
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 address = {New York},
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@article{lustro2019onset,
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 year = {2019}
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@article{lyons2013bayes,
 abstract = { In almost every scientific field, an experiment involves
collecting data and then analysing it. The analysis stage will often
consist in trying to extract some physical parameter and estimating
its uncertainty; this is known as Parameter Determination. An example
would be the determination of the mass of the top quark, from data
collected from high energy proton--proton collisions. A different aim
is to choose between two possible hypotheses. For example, are data
on the recession speed s of distant galaxies proportional to their
distance d, or do they fit better to a model where the expansion of
the Universe is accelerating? There are two fundamental approaches to
such statistical analyses -- Bayesian and Frequentist. This article
discusses the way they differ in their approach to probability, and
then goes on to consider how this affects the way they deal with
Parameter Determination and Hypothesis Testing. The examples are
taken from everyday life and from particle physics. },
 author = {Lyons, L.},
 doi = {10.1080/00107514.2012.756312},
 journal = {Contemporary Physics},
 pages = {1--16},
 title = {{B}ayes and {F}requentism: a particle physicist\textquotesingle          s perspective},
 volume = {54},
 year = {2013}
}

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 author = {X. Ma and G.E. Karniadakis},
 journal = {J. Fluid Mech.},
 pages = {181--190},
 title = {{A} low-dimensional model for simulating 3-dimensional cylinder flow},
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 author = {R. MacPherson and B. Schweinhart},
 doi = {10.1063/1.4737391},
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 title = {{M}easuring shape with topology},
 volume = {53},
 year = {2012}
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@article{magere2000simulation,
 author = {E. Mag{\'e}re and M.O. Deville},
 doi = {10.1016/S0168-9274(99)00089-6},
 journal = {Appl. Num. Math.},
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 year = {2000}
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 journal = {Science},
 publisher = {American Association for the Advancement of Science},
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 year = {2020}
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@misc{mainierinoyearnotitle,
 author = {R. Mainieri},
 note = {Appendix \textbackslash          em \textasciigrave          \textasciigrave          A brief history of chaos\textquotesingle          \textquotesingle          , in efrefD{\aa}sBuch}
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 year = {2004}
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@unpublished{manneville2014on,
 archiveprefix = {arXiv},
 author = {Manneville, P.},
 eprint = {1403.6374},
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 year = {2014}
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@article{manneville2016transition,
 archiveprefix = {arXiv},
 author = {P. Manneville},
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 eprint = {1604.00840},
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 pages = {15-00684-15-00684},
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 title = {{T}ransition to turbulence in wall-bounded flows: {W}here do we stand?},
 volume = {3},
 year = {2016}
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 author = {Manneville, P.},
 journal = {Mech. Eng. Rev.},
 pages = {15--00684},
 publisher = {The Japan Society of Mechanical Engineers},
 title = {{T}ransition to turbulence in wall-bounded flows: {W}here do we stand?},
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 year = {1984}
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@unpublished{marensi2020symmetryreduced,
 author = {E. Marensi and G. Yaln{\i}z and B. Hof and N. B. Budanur},
 note = {In preparation},
 title = {{S}ymmetry-reduced dynamic mode decomposition of the near-wall turbulence},
 year = {2020}
}

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 journal = {Physica D},
 pages = {247--276},
 title = {{B}ifurcations in systems with {Z}2 spatio-temporal and {O}(2) spatial symmetry},
 volume = {189},
 year = {2004}
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@article{marques2006onset,
 author = {F. Marques and J. M. Lopez},
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 journal = {J. Fluid Mech.},
 pages = {255--277},
 title = {{O}nset of three-dimensional unsteady states in small-aspect-ratio {T}aylor-{C}ouette flow},
 volume = {561},
 year = {2006}
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@book{marsden1976hopf,
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@book{marsden1992lectures,
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 year = {1992}
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@book{marsden1999introduction,
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 year = {1999}
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@article{martens1987local,
 abstract = {local frequency analysis is used and action-angle
coordinate is parametrized by the frequency ratio. The transport
zones and partial barriers are identified according to the },
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@article{martine1989nonlinear,
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 year = {1989}
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@article{martinerie1998epileptic,
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@article{marwan2007recurrence,
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 volume = {438},
 year = {2007}
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 title = {{A} historical review of recurrence plots},
 volume = {164},
 year = {2008}
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@book{mathews1970mathematical,
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@article{matkowsky1993stability,
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 year = {1993}
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@article{may1976simple,
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 year = {1976}
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@article{mccomb2001conditional,
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 title = {{C}onditional mode-elimination and the subgrid-modeling problem for isotropic turbulence},
 volume = {13},
 year = {2001}
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@article{mccomb2003twopoint,
 author = {W.D. McComb and A.P. Quinn},
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 journal = {Physica A},
 number = {3-4},
 pages = {487--508},
 title = {{T}wo-point, two-time closures applied to forced isotropic turbulence},
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@article{mccomb2014spectral,
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 journal = {Phys. Rev. E},
 numpages = {12},
 pages = {053010},
 publisher = {American Physical Society},
 title = {{S}pectral analysis of structure functions and their scaling exponents in forced isotropic turbulence},
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 year = {2014}
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@article{mccomb2015energy,
 author = {W. D. McComb and A. Berera and  S. R. Yoffe and M. F. Linkmann},
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 issue = {4},
 journal = {Phys. Rev. E},
 numpages = {10},
 pages = {043013},
 publisher = {American Physical Society},
 title = {{E}nergy transfer and dissipation in forced isotropic turbulence},
 volume = {91},
 year = {2015}
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@article{mccord1986uniqueness,
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 year = {1986}
}

@article{mcdonald1985fractal,
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@article{mckean1972selbergtextquotesingle,
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@article{mckeon2004further,
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@book{mees1981dynamics,
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@article{mehta2005on,
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@article{mellibovsky2009transition,
 author = {F. Mellibovsky and A. Meseguer and T. M. Schneider and B. Eckhardt},
 doi = {10.1103/PhysRevLett.103.054502},
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 journal = {Phys. Rev. Lett.},
 month = {Jul},
 numpages = {4},
 pages = {054502},
 publisher = {American Physical Society},
 title = {{T}ransition in {L}ocalized {P}ipe {F}low {T}urbulence},
 url = {https://link.aps.org/doi/10.1103/PhysRevLett.103.054502},
 volume = {103},
 year = {2009}
}

@article{mellibovsky2011from,
 archiveprefix = {arXiv},
 author = {Mellibovsky, F. and Eckhardt, B.},
 eprint = {1110.1222},
 journal = {J. Fluid Mech.},
 pages = {149--190},
 title = {{F}rom travelling waves to mild chaos: {A} supercritical bifurcation cascade in pipe flow},
 volume = {709},
 year = {2011}
}

@article{mellibovsky2011takensbogdanov,
 author = {Mellibovsky, F. and Eckhardt, B.},
 journal = {J. Fluid Mech.},
 pages = {96--129},
 title = {{T}akens--{B}ogdanov bifurcation of travelling-wave solutions in pipe flow},
 volume = {670},
 year = {2011}
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@article{mellibovsky2015mechanism,
 author = {F. Mellibovsky and A. Meseguer},
 doi = {10.1017/jfm.2015.440},
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@book{meyer1992introduction,
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 abstract = {The purpose of this paper is to develop analytical methods for studyingparticle
paths in a class of three-dimensional incompressible fluid flows.
In this paper we study three-dimensionalvolume preserving vector
fields that are invariant under the action of a one-parameter symmetry
group whose infinitesimal generator is autonomous and volume-preserving.
We show that there exists a coordinate system in which the vector
field assumes a simple form. In particular, the evolution of two
of the coordinates is governed by a time-dependent, one-degree-of-freedom
Hamiltonian system with the evolution of the remaining coordinate
being governed by a first-order differential equation that depends
only on the other two coordinates and time. The new coordinates depend
only on the symmetry group of the vector field. Therefore they arefield-independent.
The coordinate transformation is constructive. If the vector field
is time-independent, then it possesses an integral of motion. Moreover,
we show that the system can be further reduced toaction-angle-angle
coordinates. These are analogous to the familiar action-angle variables
from Hamiltonian mechanics and are quite useful for perturbative
studies of the class of systems we consider. In fact, we show how
our coordinate transformation puts us in a position to apply recent
extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional,
volume-preserving maps as well as three-dimensional versions of Melnikov\textquotesingle          s
method. We discuss the integrability of the class of flows considered,
and draw an analogy with Clebsch variables in fluid mechanics.},
 author = {I. Mezi{\' c} and S. Wiggins},
 journal = {J. Nonlin. Sci.},
 pages = {157--194},
 title = {{O}n the integrability and perturbation of three-dimensional fluid flows with symmetry},
 volume = {4},
 year = {1994}
}

@article{mezic2005spectral,
 author = {Mezi{\' c}, Igor},
 doi = {10.1007/s11071-005-2824-x},
 journal = {Nonlinear Dyn.},
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 title = {{S}pectral properties of dynamical systems, model reduction and decompositions},
 volume = {41},
 year = {2005}
}

@article{michelson1977nonlinear,
 abstract = {Consider the effect of hydrodynamics instability and diffusion thermal instability through the numerical calculation of the KSe.},
 author = {D. M. Michelson and G. I. Sivashinsky},
 journal = {Acta Astronaut.},
 pages = {1207--1221},
 title = {{N}onlinear analysis of hydrodynamic instability in laminar flames - {I}I. {N}umerical experiments},
 volume = {4},
 year = {1977}
}

@article{michelson1986steady,
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 pages = {89--111},
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 year = {1986}
}

@article{milewski2015faraday,
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 pages = {361--388},
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 volume = {778},
 year = {2015}
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@inproceedings{milnor1988on,
 author = {J. Milnor and W. Thurston},
 booktitle = {Dynamical Systems},
 editor = {J. C. Alexander},
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 publisher = {Springer Berlin Heidelberg},
 title = {{O}n iterated maps of the interval},
 year = {1988}
}

@misc{minic2011on,
 archiveprefix = {arXiv},
 author = {Mini{\' c}, Dj. and Pleimling, M. and Staples, A. E.},
 eprint = {1105.2941},
 title = {{O}n the steady state distributions for turbulence},
 year = {2011}
}

@article{miranda1993protolorenz,
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 volume = {178},
 year = {1993}
}

@article{mischaikow1998chaos,
 author = {K. Mischaikow and M. Mrozek},
 journal = {Math. Comp.},
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 year = {1998}
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@article{mischaikow1999construction,
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 year = {1999}
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@article{moehlis2004fractal,
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 year = {2004}
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@article{moehlis2005periodic,
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 year = {2005}
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@article{moffatt1969degree,
 author = {Moffatt, H. K.},
 journal = {J. Fluid Mech},
 pages = {117--129},
 title = {{T}he degree of knottedness of tangled vortex lines},
 volume = {35},
 year = {1969}
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@inproceedings{moffatt1990fixed,
 address = {New York},
 author = {H. K. Moffatt},
 booktitle = {Whither Turbulence? Turbulence at the Crossroads},
 editor = {J. L. Lumley},
 pages = {250--257},
 publisher = {Springer},
 title = {{F}ixed points of turbulent dynamical systems and suppression of nonlinearity, {C}omment 1},
 year = {1990}
}

@article{mohan2017scaling,
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 doi = {10.1103/physrevfluids.2.114606},
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 month = {nov},
 number = {11},
 publisher = {American Physical Society (APS)},
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 year = {2017}
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@article{moin1998direct,
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 year = {1998}
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@article{molavcek2013drops,
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 journal = {J. Fluid Mech.},
 pages = {582--611},
 publisher = {Cambridge University Press},
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 volume = {727},
 year = {2013}
}

@article{molavcek2013dropsb,
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 year = {2013}
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@book{monin1971statistical,
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 title = {{S}tatistical {F}luid {M}echanics},
 year = {1971}
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 abstract = {There exist a band of winding numbers associated with
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winding numbers decay to the ones with smaller winding numbers.
Defect chaos happens when the range of stable winding numbers
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 volume = {325},
 year = {1988}
}

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 journal = {J. Nonlinear Science},
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 title = {{R}elative equilibria of molecules},
 volume = {9},
 year = {1999}
}

@book{moon1987chaotic,
 address = {New York},
 author = {F. C. Moon},
 publisher = {Wiley},
 title = {{C}haotic {V}ibrations: {A}n {I}ntroduction for {A}pplied {S}cientists and {E}ngineers},
 year = {1987}
}

@article{moore1966thermally,
 author = {D. W. Moore and E. A. Spiegel},
 journal = {Astrophys. J.},
 pages = {871},
 title = {{A} thermally excited nonlinear oscillator},
 volume = {143},
 year = {1966}
}

@article{moresco1999pimsimplex,
 abstract = {They generalized the PIM method to treat the chaotic
saddle with more than one unstable direction by introducing a
simplex to cross the stable manifolds. Combining with one method for
finding local extremum in a simplex, the authors do essentially the
same job as in the simple PIM method.},
 author = {P. Moresco and S. P. Dawson},
 journal = {Physica D},
 pages = {38},
 title = {{T}he {P}IM-simplex method: an extension of the {P}IM-triple method to saddles with an arbitrary number of expanding directions},
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 year = {1999}
}

@article{morita2010scytale,
 abstract = {A method for estimating a period of unstable periodic solutions
is suggested in continuous dissipative chaotic dynamical
systems. The measurement of a minimum distance between a
reference state and an image of transformation of it exhibits a
characteristic structure of the system, and the local minima of
the structure give candidates of period and state of
corresponding symmetric solutions. Appropriate periods and
initial states for the Newton method are chosen efficiently by
setting a threshold to the range of the minimum distance and
the period.},
 author = {Morita, Y. and Fujiwara, N. and Kobayashi, M. U. and Mizuguchi, T.},
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 year = {2010}
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@article{morrison1980maxwellvlasov,
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 volume = {80},
 year = {1980}
}

@article{morrison1980noncanonical,
 author = {Morrison, P. J. and J. M. Greene},
 journal = {Phys. Rev. Lett.},
 note = {see also Phys. Rev. Lett. 48, 569 (1982)},
 pages = {790--794},
 title = {{N}oncanonical {H}amiltonian {D}ensity {F}ormulation of {H}ydrodynamics and {I}deal {M}agnetohydrodynamics},
 volume = {45},
 year = {1980}
}

@article{morrison1984scattering,
 abstract = {The Lagrangian density for the regularized-long-wave equation
(also known as the BBM equation) is presented. Using the trial
function technique, ordinary differential equations that describe
the time dependence of the position of the peaks, amplitudes,
and widths for the collision of two solitary waves are obtained.
These equations are analyzed in the Born and \textasciigrave          \textasciigrave          equal-width\textquotesingle          \textquotesingle          
approximations and compared with numerical results obtained
by direct integration utilizing the split-step fast Fourier-transform
method. The computations show that collisions are inelastic and
that production of solitary waves may occur.},
 author = {P. J. Morrison and J. D. Meiss and J. R. Cary},
 journal = {Physica D},
 pages = {324--336},
 title = {{S}cattering of regularized-long-wave solitary waves},
 volume = {11},
 year = {1984}
}

@article{morrison1998hamiltonian,
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 volume = {70},
 year = {1998}
}

@article{morrison2002reynolds,
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 title = {{R}eynolds number dependence of streamwise velocity spectra in turbulent pipe flow},
 volume = {88},
 year = {2002}
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@article{morrison2004scaling,
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@article{morriss1994notitle,
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 volume = {75},
 year = {1994}
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@article{moser1966rapidly,
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 journal = {Ann. Scuola Norm. Super. Pisa},
 pages = {499--535},
 title = {{A} rapidly convergent iteration method and non-linear differential equations. {I}I},
 url = {http://www.numdam.org/numdam-bin/item?id=ASNSP\_1966\_3\_20\_3\_499\_0},
 volume = {20},
 year = {1966}
}

@book{moser1973stable,
 address = {Princeton},
 author = {J. Moser},
 publisher = {Princeton Univ. Press},
 title = {{S}table and {R}andom {M}otions in {D}ynamical {S}ystems},
 year = {1973}
}

@article{moser1983spectral,
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 journal = {J. Comp. Phys.},
 pages = {524--544},
 title = {{A} spectral method for the {N}avier-{S}tokes equations with applications to {T}aylor-{C}ouette flow},
 volume = {52},
 year = {1983}
}

@article{moser1999direct,
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 journal = {Phys. Fluids},
 pages = {943--946},
 title = {{D}irect numerical simulation of turbulent channel flow up to \textdollar          {R}e\_\tau           = 590\textdollar          },
 volume = {11},
 year = {1999}
}

@article{mostow1957equivariant,
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@article{mowlavi2018model,
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@book{mullin2005nonuniqueness,
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 publisher = {Kluwer},
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@incollection{mullin2006recent,
 address = {New York},
 author = {T. Mullin and J. Peixinho},
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 editor = {Govindarajan, R.},
 pages = {45--55},
 publisher = {Springer},
 series = {Fluid Mechanics and Its Applications},
 title = {{R}ecent observations of the transition to turbulence in a pipe},
 volume = {78},
 year = {2006}
}

@article{mullin2011experimental,
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 year = {2011}
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@article{myers2019persistent,
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@article{nagata1983threedimensional,
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 year = {1983}
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@article{nagata1990threedimensional,
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 year = {1990}
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@article{nagata1996nonlinear,
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@article{nagata1997threedimensional,
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 year = {1997}
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@inproceedings{nagata2004threedimensional,
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@article{nagata2013mirrorsymmetric,
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 year = {2013}
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@article{nakamura1996existence,
 author = {Nakamura, I. and Toya, Y.},
 doi = {10.1007/BF01181035},
 journal = {Acta Mechanica},
 pages = {33--46},
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 year = {1996}
}

@article{nambu1973generalized,
 author = {Nambu, Y.},
 doi = {10.1103/PhysRevD.7.2405},
 journal = {Phys. Rev. D},
 pages = {2405--2414},
 title = {{G}eneralized {H}amiltonian dynamics},
 volume = {7},
 year = {1973}
}

@incollection{narasimha1989utility,
 address = {Berlin},
 author = {R. Narasimha},
 booktitle = {Whither Turbulence? Turbulence at the Cross-Road},
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 pages = {13--48},
 publisher = {Springer},
 title = {{T}he utility and drawbacks of traditional approaches},
 year = {1989}
}

@article{navier1823memoire,
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 journal = {M{\'e}moires de l\textquotesingle          Acad{\'e}mie Royale des Sciences de l\textquotesingle          Institut de France},
 pages = {389--440},
 title = {{M}{\'e}moire sur les lois du mouvement des fluides},
 volume = {6},
 year = {1823}
}

@article{neimark1959on,
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 year = {1959}
}

@article{nelkin1992in,
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 journal = {Science},
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 pages = {566--570},
 publisher = {American Association for the Advancement of Science},
 title = {{I}n what sense is turbulence an unsolved problem?},
 volume = {255},
 year = {1992}
}

@article{newell1974envelope,
 author = {A. C. Newell},
 journal = {Lect. Appl. Math.},
 pages = {157--163},
 title = {{E}nvelope equations},
 volume = {15},
 year = {1974}
}

@article{newhouse1978occurrence,
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 journal = {Commun. Math. Phys.},
 pages = {35},
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 volume = {64},
 year = {1978}
}

@misc{news2020coronavirus,
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@misc{news2020qa,
 author = {BBC News},
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@article{newton1687philosophiae,
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 year = {1687}
}

@article{newton1986instabilities,
 author = {P. K. Newton and L. Sirovich},
 journal = {Quart. Appl. Math.},
 pages = {49--58},
 title = {{I}nstabilities of {G}inzburg-{L}andau equation: periodic solutions},
 volume = {XLIV},
 year = {1986}
}

@article{newton1986instabilitiesb,
 author = {P. K. Newton and L. Sirovich},
 journal = {Quart. Appl. Math.},
 pages = {367--374},
 title = {{I}nstabilities of {G}inzburg-{L}andau equation: secondary bifurcation},
 volume = {XLIV},
 year = {1986}
}

@article{nicolaenko1985some,
 abstract = {The KSe shows coherent spatial patterns but temporal chaos. The anti-diffusion term destablizes the solutin while stability is achived by the collective action of all modes through the nonlinear convection term. An estimate of the L\^{}2 norm of the solution in the antisymmetric periodic function space is given. The number of determining Fourier modes is calculated together with fractal dimension of the strange attractor.},
 author = {B. Nicolaenko and B. Scheurer and R. Temam},
 journal = {Physica D},
 pages = {155--183},
 title = {{S}ome global dynamical properties of the {K}uramoto-{S}ivashinsky equations: nonlinear stability and attractors},
 volume = {16},
 year = {1985}
}

@article{ning1990detuned,
 author = {Ning, C. Z. and Haken, H.},
 doi = {10.1103/PhysRevA.41.3826},
 journal = {Phys. Rev.},
 pages = {3826},
 title = {{D}etuned lasers and the complex {L}orenz equations: {S}ubcritical and supercritical {H}opf bifurcations},
 volume = {41},
 year = {1990}
}

@article{ning1991phase,
 author = {Ning, C. Z. and Haken, H.},
 doi = {10.1103/PhysRevA.43.6410},
 journal = {Phys. Rev. A},
 pages = {6410--6413},
 title = {{P}hase anholonomy in dissipative optical-systems with periodic oscillations},
 volume = {43},
 year = {1991}
}

@article{ning1992geometrical,
 abstract = {We show that the geometrical (Berry) phases discovered in Hamiltonian
systems can also be defined as resulting from parallel transportation
of vectors for nonlinear dissipative systems with cyclic attractors.
If the nonlinear dissipative systems possess a certain kind of asymptotic
solution defined in this Letter, the phase and amplitude accumulation
of a geometrical type can be defined. Detuned one- and two-photon
lasers showing periodic intensity pulsations are taken as examples
of such systems.},
 author = {C. Z. Ning and H. Haken},
 journal = {Phys. Rev. Lett.},
 pages = {2109--2112},
 title = {{G}eometrical phase and amplitude accumulations in dissipative systems with cyclic attractors},
 volume = {68},
 year = {1992}
}

@article{ning1992invariance,
 abstract = {We show that the geometrical phase defined for the dissipative systems
[1] is invariant under a unitary transformation. This implies for
lasers an invariance of the geometrical phase for different choices
of the reference frequency.},
 author = {Ning, C. Z. and Haken, H.},
 doi = {10.1007/BF01318153},
 journal = {Z. Phys. B},
 pages = {261--262},
 title = {{A}n invariance property of the geometrical phase and its consequence in detuned lasers},
 volume = {89},
 year = {1992}
}

@article{nishi2008laminartoturbulent,
 author = {Nishi, M. and B{\"u}lent, {\"U}. and Durst, F. and Biswas, G.},
 journal = {J. Fluid Mech.},
 pages = {425--446},
 title = {{L}aminar-to-turbulent transition of pipe flows through puffs and slugs},
 volume = {614},
 year = {2008}
}

@article{nizette2001stability,
 author = {M. Nizette and T. Erneux and A. Gavrielides and V. Kovanis},
 journal = {Phys. Rev. E},
 pages = {026212},
 title = {{S}tability and bifurcations of periodically modulated, optically injected laser diodes},
 volume = {63},
 year = {2001}
}

@article{noack2003hierarchy,
 author = {B. R. Noack and K. Afanasiev and M. Morzynski and F. Thiele},
 journal = {J. Fluid Mech.},
 pages = {335--363},
 title = {{A} hierarchy of low-dimensional models for the transient and post-transient cylinder wake},
 volume = {497},
 year = {2003}
}

@book{noack2011reducedorder,
 author = {B. R. Noack and M. Morzynski and G. Tadmor},
 publisher = {Springer Science \& Business Media},
 title = {{R}educed-order modelling for flow control},
 volume = {528},
 year = {2011}
}

@article{noack2016recursive,
 author = {Noack, Be. R. and Stankiewicz, W. and Morzy\textbackslash          'nski, M. and Schmid, P. J.},
 doi = {10.1017/jfm.2016.678},
 journal = {J. Fluid Mech.},
 pages = {843--872},
 publisher = {Cambridge University Press},
 title = {{R}ecursive dynamic mode decomposition of transient and post-transient wake flows},
 volume = {809},
 year = {2016}
}

@article{noether1915der,
 author = {Noether, E.},
 doi = {10.1007/BF01456821},
 journal = {Math. Ann.},
 pages = {89--92},
 title = {{D}er {E}ndlichkeitssatz der {I}nvarianten endlicher {G}ruppen},
 volume = {77},
 year = {1915}
}

@article{nozaki1983pattern,
 author = {K. Nozaki and N. Bekki},
 journal = {Phys. Rev. Lett.},
 pages = {2171},
 title = {{P}attern selection and spatiotemporal transition to chaos in the {G}inzburg-{L}andau equation},
 volume = {51},
 year = {1983}
}

@article{nozaki1984exact,
 author = {K. Nozaki and N. Bekki},
 journal = {J. Phys. Soc. Japan},
 pages = {1581--1582},
 title = {{E}xact {S}olutions of the {G}eneralized {G}inzburg-{L}andau {E}quation},
 volume = {53},
 year = {1984}
}

@article{nusse1989procedure,
 abstract = {PIM (proper interior maximum) method is proposed to
find chaotic trajectories on a nonattracting set with one-d unstable
manifold. By partitioning a straight line straddling the stable
manifolds, the point with longer and longer escaping time is found
in a sequel until a sufficiently small interval is reached when the
point is considered to be very close to the stable manifolds. After
that, evolving and restraining the line element results in a
NUMERICAL orbit is obtained.},
 author = {H. E. Nusse and J. A. Yorke},
 journal = {Physica D},
 pages = {137},
 title = {{A} procedure for finding numerical trajectories on chaotic saddles},
 volume = {36},
 year = {1989}
}

@unpublished{olesen2007exact,
 abstract = {derive exact analytical solutions of the GOY shell model
of turbulence. In the absence of forcing and viscosity obtain
closed form solutions in terms of Jacobi elliptic functions. With
three shells the model is integrable. Obtain a Kolmogorov solution
in the limit of infinitely many shells. Also identify cases where
the models exhibits transitions to chaotic states lying on strange
attractors or ergodic energy surfaces.},
 archiveprefix = {arXiv},
 author = {Olesen, P. and Jensen, M.s H.},
 eprint = {0705.3123},
 title = {{E}xact periodic solutions of shells models of turbulence},
 year = {2007}
}

@book{olver1998applications,
 address = {New York},
 author = {P. J. Olver},
 publisher = {Spring-Verlag},
 series = {GTM},
 title = {{A}pplications of {L}ie {G}roups to {D}ifferential {E}quations},
 volume = {107},
 year = {1998}
}

@book{olver1999classical,
 address = {Cambridge},
 author = {P. J. Olver},
 publisher = {Cambridge Univ. Press},
 title = {{C}lassical {I}nvariant {T}heory},
 year = {1999}
}

@book{olver2006applied,
 address = {Upper Saddle River, NJ},
 author = {Olver, P. J. and Shakiban, C.},
 publisher = {Prentice-Hall},
 title = {{A}pplied {L}inear {A}lgebra},
 year = {2006}
}

@article{onsager1953fluctuations,
 author = {L. Onsager and S. Machlup},
 doi = {10.1103/PhysRev.91.1505},
 journal = {Phys. Rev.},
 pages = {1505--1512},
 title = {{F}luctuations and irreversible processes},
 volume = {91},
 year = {1953}
}

@article{oono1980chaos,
 author = {Y. Oono and Y. Takahashi},
 doi = {10.1143/PTP.63.1804},
 journal = {Progr. Theor. Phys.},
 pages = {1804--1807},
 title = {{C}haos, external noise and {F}redholm theory},
 volume = {63},
 year = {1980}
}

@article{orr1907stability,
 author = {Orr, W. M. F.},
 journal = {Proc. Roy. Irish Acad. A},
 pages = {69--138},
 title = {{T}he stability or instability of the steady motions of a liquid. {P}art {I}I: {A} viscous liquid},
 volume = {27},
 year = {1907}
}

@article{orszag1969numerical,
 author = {S. A. Orszag},
 doi = {10.1063/1.1692445},
 journal = {Phys. Fluids},
 number = {12},
 pages = {II-250-II-257},
 title = {{N}umerical {M}ethods for the {S}imulation of {T}urbulence},
 volume = {12},
 year = {1969}
}

@article{orszag1971accurate,
 abstract = {The Orr-Sommerfeld equation is solved numerically using
expansions in Chebyshev polynomials and the QR matrix eigenvalue
algorithm. It is shown that results of great accuracy are obtained
very economically. The method is applied to the stability of plane
Poiseuille flow. Expansions in Chebyshev polynomials are better
suited to the solution of hydrodynamic stability problems than
expansions in other, seemingly more relevant, sets of orthogonal
functions.},
 author = {S. A. Orszag},
 journal = {J. Fluid Mech.},
 pages = {689--703},
 title = {{A}ccurate solution of the {O}rr-{S}ommerfeld stability equation},
 volume = {50},
 year = {1971}
}

@article{orszag1980transition,
 author = {S. A. Orszag and L. C. Kells},
 journal = {J. Fluid Mech.},
 pages = {159--205},
 title = {{T}ransition to turbulence in plane {P}oiseuille and plane {C}ouette flow},
 volume = {96},
 year = {1980}
}

@article{orszag1986boundary,
 author = {S. A. Orszag and M. Israeli and M. Deville},
 journal = {J. Sci. Comput.},
 pages = {75--111},
 title = {{B}oundary conditions for incompressible flows},
 volume = {1},
 year = {1986}
}

@article{oseledec1968multiplicative,
 abstract = {The Liapunov characteristic numbers are defined by investigating the limiting volume of infinitesimal size. Their existence is proved by construction and application of the ergodic theorem. At a.e. every point, the tangent space is divided into stable and unstable subspace corresponding the positive and negative Liapunov numbers. If the eigenvalues are non-degenerate, the Liyapunov vectors are well defined a.e.},
 author = {V. I. Oseledec},
 journal = {Trans. Moscow Math. Soc.},
 pages = {197--221},
 title = {{A} multiplicative ergodic theorem. {L}iapunov characteristic numbers for dynamical systems},
 volume = {19},
 year = {1968}
}

@article{ostrowski1962some,
 author = {A. Ostrowski and H. Schneider},
 journal = {J. Math. Anal. Appl.},
 pages = {72--84},
 title = {{S}ome theorems on the inertia of general matrices},
 volume = {4},
 year = {1962}
}

@article{ott1990controlling,
 author = {E. Ott and C. Grebogi and J. A. Yorke},
 journal = {Phys. Rev. Lett},
 pages = {1196},
 title = {{C}ontrolling chaos},
 volume = {64},
 year = {1990}
}

@book{ott2002chaos,
 author = {E. Ott},
 doi = {10.1017/CBO9780511803260},
 edition = {2nd},
 place = {Cambridge},
 publisher = {Cambridge University Press},
 title = {{C}haos in {D}ynamical {S}ystems},
 year = {2002}
}

@inbook{ott2002chaotic,
 author = {Ott, Edward},
 booktitle = {Chaos in Dynamical Systems},
 doi = {10.1017/CBO9780511803260.010},
 edition = {2nd},
 pages = {304--344},
 place = {Cambridge},
 publisher = {Cambridge University Press},
 title = {{C}haotic transitions},
 year = {2002}
}

@inbook{ott2002nonattracting,
 author = {E. Ott},
 booktitle = {Chaos in Dynamical Systems},
 doi = {10.1017/CBO9780511803260.007},
 edition = {2nd},
 pages = {168--211},
 place = {Cambridge},
 publisher = {Cambridge University Press},
 title = {{N}onattracting chaotic sets},
 year = {2002}
}

@article{otter2017roadmap,
 archiveprefix = {arXiv},
 author = {N. Otter and M. A. Porter and U. Tillmann and P. Grindrod and H. A. Harrington},
 doi = {10.1140/epjds/s13688-017-0109-5},
 eprint = {1506.08903},
 journal = {EPJ Data Sci},
 number = {1},
 pages = {17},
 primaryclass = {math.AT},
 title = {{A} roadmap for the computation of persistent homology},
 volume = {6},
 year = {2017}
}

@article{oza2013trajectory,
 author = {A. U. Oza and R. R. Rosales and J. W. M. Bush},
 journal = {J. Fluid Mech.},
 pages = {552--570},
 publisher = {Cambridge University Press},
 title = {{A} trajectory equation for walking droplets: hydrodynamic pilot-wave theory},
 volume = {737},
 year = {2013}
}

@article{oza2014pilotwave,
 author = {A. U. Oza and  {\O}. Wind-Willassen  and D. M. Harris and R. R. Rosales and J. W. M. Bush},
 doi = {10.1063/1.4891568},
 journal = {Phys. Fluids},
 number = {8},
 pages = {082101},
 title = {{P}ilot-wave hydrodynamics in a rotating frame: {E}xotic orbits},
 volume = {26},
 year = {2014}
}

@article{page2019koopman,
 archiveprefix = {arXiv},
 author = {Page, J. and Kerswell, R.},
 doi = {10.1017/jfm.2019.686},
 eprint = {1811.05907},
 journal = {J. Fluid Mech.},
 pages = {1--27},
 publisher = {Cambridge University Press},
 title = {{K}oopman mode expansions between simple invariant solutions},
 volume = {879},
 year = {2019}
}

@misc{page2019searching,
 author = {Page, J. and Kerswell, R. R.},
 doi = {https://arxiv.org/abs/1906.01310},
 title = {{S}earching turbulence for periodic orbits with dynamic mode decomposition},
 year = {2019}
}

@article{page2020searching,
 archiveprefix = {arXiv},
 author = {J. Page and R. R. Kerswell},
 doi = {10.1017/jfm.2019.1074},
 eprint = {1906.01310},
 journal = {J. Fluid Mech.},
 pages = {A28},
 title = {{S}earching turbulence for periodic orbits with dynamic mode decomposition},
 volume = {886},
 year = {2020}
}

@article{palacios2007heteroclinic,
 author = {A. Palacios},
 journal = {Scholarpedia},
 pages = {2352},
 title = {{H}eteroclinic cycles},
 volume = {2},
 year = {2007}
}

@article{palais1961on,
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 journal = {Ann. Math.},
 pages = {295--323},
 title = {{O}n the existence of slices for actions of non-compact {L}ie groups},
 volume = {73},
 year = {1961}
}

@book{panton1996incompressible,
 address = {New York},
 author = {R. L. Panton},
 publisher = {Wiley},
 title = {{I}ncompressible flow},
 year = {1996}
}

@book{panton1997selfsustaining,
 address = {Southhampton},
 editor = {R. L. Panton},
 publisher = {Computational Mechanics Publications},
 title = {{S}elf-sustaining mechanisms of wall turbulence},
 year = {1997}
}

@book{papoulis2002probability,
 author = {A. Papoulis and S. U. Pillai},
 isbn = {9780070486584},
 publisher = {McGraw-Hill},
 title = {{P}robability, {R}andom {V}ariables, and {S}tochastic {P}rocesses},
 year = {2002}
}

@article{paranjape2020oblique,
 author = {C. S. Paranjape and Y. Duguet and B. Hof},
 doi = {10.1017/jfm.2020.322},
 journal = {J. Fluid Mech.},
 pages = {A7},
 publisher = {Cambridge University Press},
 title = {{O}blique stripe solutions of channel flow},
 volume = {897},
 year = {2020}
}

@article{parlitz1986resonances,
 author = {U. Parlitz and W. Lauterborn},
 journal = {Z. Naturforsch. A},
 number = {4},
 pages = {605--614},
 title = {{R}esonances and torsion numbers of driven dissipative nonlinear oscillators},
 volume = {41},
 year = {1986}
}

@article{parlitz1987perioddoubling,
 author = {U. Parlitz and W. Lauterborn},
 journal = {Phys. Rev. A},
 pages = {1428},
 title = {{P}eriod-doubling cascades and devil\textquotesingle          s staircases of the driven van der {P}ol oscillator},
 volume = {36},
 year = {1987}
}

@article{parry1983analogue,
 author = {W. Parry and M. Pollicott},
 journal = {Ann. Math.},
 pages = {573--591},
 title = {{A}n analogue of the prime number theorem for closed orbits of {A}xiom {A} flows},
 volume = {118},
 year = {1983}
}

@article{parry1990zeta,
 author = {W. Parry and M. Pollicott},
 journal = {Ast{\'e}risque},
 pages = {268},
 title = {{Z}eta functions and the periodic orbit structure of hyperbolic dynamics},
 volume = {187-188},
 year = {1990}
}

@article{pasquarelli1987spectral,
 author = {F. Pasquarelli and A. Quarteroni and G. Sacchi-Landriani},
 journal = {J. Comp. Sci.},
 pages = {195--226},
 title = {{S}pectral approximations of the {S}tokes problem by divergence-free functions},
 volume = {2},
 year = {1987}
}

@article{pathak2017using,
 archiveprefix = {arXiv},
 author = {J. Pathak  and Z. Lu  and B. R. Hunt  and M. Girvan  and E. Ott},
 doi = {10.1063/1.5010300},
 eprint = {1710.07313},
 journal = {Chaos},
 number = {12},
 pages = {121102},
 title = {{U}sing machine learning to replicate chaotic attractors and calculate {L}yapunov exponents from data},
 volume = {27},
 year = {2017}
}

@article{pathak2018hybrid,
 archiveprefix = {arXiv},
 author = {J. Pathak  and A. Wikner  and R. Fussell  and S. Chandra  and B. R. Hunt  and M. Girvan and E. Ott},
 doi = {10.1063/1.5028373},
 eprint = {1803.04779},
 journal = {Chaos},
 number = {4},
 pages = {041101},
 title = {{H}ybrid forecasting of chaotic processes: {U}sing machine learning in conjunction with a knowledge-based model},
 volume = {28},
 year = {2018}
}

@article{pathak2018modelfree,
 author = {J. Pathak and B. Hunt and M. Girvan and Z. Lu and E. Ott},
 doi = {10.1103/PhysRevLett.120.024102},
 issue = {2},
 journal = {Phys. Rev. Lett.},
 month = {Jan},
 numpages = {5},
 pages = {024102},
 publisher = {American Physical Society},
 title = {{M}odel-{F}ree {P}rediction of {L}arge {S}patiotemporally {C}haotic {S}ystems from {D}ata: {A} Reservoir {C}omputing {A}pproach},
 volume = {120},
 year = {2018}
}

@article{peacock2013lagrangian,
 author = {T. Peacock and G. Haller},
 bdsk-url-1 = {http://dx.doi.org/10.1063/PT.3.1886},
 doi = {10.1063/PT.3.1886},
 journal = {Physics Today},
 number = {2},
 pages = {41--47},
 title = {{L}agrangian coherent structures: {T}he hidden skeleton of fluid flows},
 volume = {66},
 year = {2013}
}

@article{peckham1990necessity,
 author = {Peckham, B. B.},
 journal = {Nonlinearity},
 pages = {261},
 title = {{T}he necessity of the {H}opf bifurcation for periodically forced oscillators},
 volume = {3},
 year = {1990}
}

@article{peixinho2006decay,
 author = {J. Peixinho and T. Mullin},
 journal = {Phys. Rev. Lett.},
 pages = {094501},
 title = {{D}ecay of turbulence in pipe flow},
 volume = {96},
 year = {2006}
}

@article{peixinho2007finiteamplitude,
 author = {J. Peixinho and T. Mullin},
 journal = {J. Fluid Mech.},
 pages = {169--178},
 title = {{F}inite-amplitude thresholds for transition in pipe flow},
 volume = {582},
 year = {2007}
}

@article{peng2007potentialflow,
 author = {J. Peng and J. O. Dabiri},
 doi = {10.1007/s00348-007-0315-1},
 issn = {1432-1114},
 journal = {Exp. Fluids},
 pages = {655--664},
 title = {{A} potential-flow, deformable-body model for fluid-structure interactions with compact vorticity: application to animal swimming measurements},
 volume = {43},
 year = {2007}
}

@book{penrose2005road,
 address = {New York},
 author = {R. Penrose},
 publisher = {A. A. Knopf},
 title = {{T}he {R}oad to {R}eality: {A} Complete {G}uide to the {L}aws of the {U}niverse},
 year = {2005}
}

@article{perea2015sliding,
 archiveprefix = {arXiv},
 author = {J. A. Perea and J. Harer},
 eprint = {1307.6188},
 journal = {Found. Comput. Math.},
 number = {3},
 pages = {799--838},
 publisher = {Springer},
 title = {{S}liding windows and persistence: {A}n application of topological methods to signal analysis},
 volume = {15},
 year = {2015}
}

@article{pereira1977nonlinear,
 abstract = {A nonlinear Schr{\"o}dinger equation is found with growth and damping terms in the weakly nonlinear regime. An exact solitary solution is found for the equation.},
 author = {N. R. Pereira and L. Stenflo},
 journal = {Phys. Fluid},
 pages = {1733--1734},
 title = {{N}onlinear {S}chr{\"o}dinger equation including growth and damping},
 volume = {20},
 year = {1977}
}

@book{perko1991differential,
 address = {New York},
 author = {L. Perko},
 publisher = {Springer},
 title = {{D}ifferential {E}quations and {D}ynamical {S}ystems},
 year = {1991}
}

@article{perrard2014chaos,
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 journal = {Phys. Rev. Lett.},
 number = {10},
 pages = {104101},
 publisher = {American Physical Society},
 title = {{C}haos driven by interfering memory},
 volume = {113},
 year = {2014}
}

@article{perrard2014selforganization,
 author = {S. Perrard and M. Labousse and M. Miskin and E. Fort and Y. Couder},
 journal = {Nature communications},
 publisher = {Nature Publishing Group},
 title = {{S}elf-organization into quantized eigenstates of a classical wave-driven particle},
 volume = {5},
 year = {2014}
}

@article{perrard2018transition,
 author = {S. Perrard and M. Labousse},
 doi = {10.1063/1.5032088},
 journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
 number = {9},
 pages = {096109},
 title = {{T}ransition to chaos in wave memory dynamics in a harmonic well: {D}eterministic and noise-driven behavior},
 volume = {28},
 year = {2018}
}

@article{peterhof1999alloptical,
 author = {D. Peterhof and B. Sandstede},
 journal = {J. Nonlinear Sci.},
 pages = {575--613},
 title = {{A}ll-optical clock recovery using multisection distributed-feedback lasers},
 volume = {9},
 year = {1999}
}

@article{pethel2006symbolic,
 author = {Pethel, S. D. and Corron, N. J. and Bollt, E.},
 doi = {10.1103/PhysRevLett.96.034105},
 journal = {Phys. Rev. Lett.},
 pages = {034105},
 title = {{S}ymbolic dynamics of coupled map lattices},
 volume = {96},
 year = {2006}
}

@article{pethel2007deconstructing,
 author = {Pethel, S. D. and Corron, N. J. and Bollt, E.},
 doi = {10.1103/PhysRevLett.99.214101},
 journal = {Phys. Rev. Lett.},
 pages = {214101},
 title = {{D}econstructing spatiotemporal chaos using local symbolic dynamics},
 volume = {99},
 year = {2007}
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@article{petit1986satellite,
 author = {J.-M. Petit and M. Henon},
 doi = {10.1016/0019-1035(86)90089-8},
 journal = {Icarus},
 number = {3},
 pages = {536 -- 555},
 title = {{S}atellite encounters},
 volume = {66},
 year = {1986}
}

@article{petrov1940application,
 author = {Petrov, G.I.},
 journal = {Prikl. Mat. Mekh.},
 pages = {3--12},
 title = {{A}pplication of {G}alerkin\textquotesingle          s method to the problem of the stability of the flow of a viscous fluid},
 volume = {4},
 year = {1940}
}

@book{peyret2002spectral,
 address = {New York},
 author = {R. Peyret},
 publisher = {Springer},
 title = {{S}pectral {M}ethods for {I}ncompressible {F}lows},
 year = {2002}
}

@incollection{pfenninger1961transition,
 address = {Oxford, UK},
 author = {W. Pfenninger},
 booktitle = {Boundary Layer and Flow Control},
 doi = {10.1016/B978-1-4832-1323-1.50013-0},
 editor = {G. V. Lachmann},
 pages = {970--980},
 publisher = {Pergamon},
 title = {{T}ransition in the inlet length of tubes at high {R}eynolds numbers},
 year = {1961}
}

@article{pfister1988bifurcation,
 author = {G. Pfister and H. Schmidt and K. A. Cliffe and T. Mullin},
 doi = {10.1017/S0022112088001491},
 journal = {J. Fluid Mech.},
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@book{poincare1992new,
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@book{pope2000turbulent,
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@misc{postlethwaite2008stabilisation,
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@article{pringle2010using,
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@article{pringle2012minimal,
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@article{pringle2015fully,
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@article{pucci2016nonspecular,
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 year = {2016}
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@article{pucci2018walking,
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@phdthesis{r2013transition,
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@article{rawat2014relative,
 abstract = {A branch of relative periodic orbits is found in plane
Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re
= 3000 to Re = 5000 . These solutions consist in sinuous quasi-streamwise
streaks periodically forced by quasi-streamwise vortices in a
self-sustained process. The streaks and the vortices are located in the
bulk of the flow. Only the amplitude, but not the shape, of the averaged
velocity components does change as the Reynolds number is increased from
3000 to 5000. We conjecture that these solutions could therefore be
related to large- and very large-scale structures observed in the bulk of
fully developed turbulent channel flows. },
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@article{rawat2015on,
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@article{recke2011frequency,
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@misc{recke2011frequencyb,
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@article{reiss2018shifted,
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@book{report2001physics,
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@book{richter2000semiclassical,
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Couette flow solutions reported by Cherhabili and
Ehrenstein [Eur. J.  Mech.  B/Fluids 14, 667 (1995)] using
their Poiseuille-Couette homotopy. We also demonstrate that
the steady solutions obtained by Mehta and Healey [Phys.
Fluids 17, 094108 (2005)] for small symmetry-preserving
perturbations are influenced by an artefact of the modified
system of equations used in their paper.},
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 author = {Rincon, F.},
 eprint = {0706.1165},
 journal = {Phys. Fluids},
 pages = {4105},
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 year = {2007}
}

@article{rincon2007onb,
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@book{risken1996fokkerplanck,
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@article{ritter2016emergence,
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@techreport{robinson1991kinematics,
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 year = {1991}
}

@article{rodriguez1990lowdimensional,
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 year = {1990}
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@article{roldan1997review,
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 doi = {10.1088/1355-5111/9/1/001},
 journal = {Quantum Semiclass. Opt.},
 pages = {R1},
 title = {{R}eview article: {T}he dynamics of optically pumped molecular lasers. {O}n its relation with the {L}orenz - {H}aken model},
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 year = {1997}
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@article{rolland2014turbulent,
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 journal = {Fluid Dyn. Res.},
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 title = {{T}urbulent spot growth in plane {C}ouette flow: statistical study and formation of spanwise vorticity},
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@article{romanov1973stability,
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@article{rome1969direct,
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@article{rosales2005linear,
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 publisher = {AIP Publishing},
 title = {{L}inear forcing in numerical simulations of isotropic turbulence: {P}hysical space implementations and convergence properties},
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 year = {2005}
}

@article{roser2020coronavirus,
 author = {M. Roser and H. Ritchie and E. Ortiz-Ospina and J. Hasell},
 journal = {Published online at OurWorldInData.org},
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 title = {{C}oronavirus {P}andemic ({C}OV{I}D-19)},
 year = {2020}
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@article{rossler1976equation,
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 year = {1976}
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@article{rosso2007distinguishing,
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 journal = {Phys. Rev. Lett.},
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@article{rothstein1999persistent,
 abstract = {Experimentally generate a 2-d periodic(or weakly
turbulent) velocity field using glycerol-water on magnets driven by
electric current. Persistent spatial patterns are observed, which
decays exponentially.},
 author = {D. Rothstein and E. Henry and J. P. Gollub},
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 title = {{P}ersistent patterns in transient chaotic fluid mixing},
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@article{roupas2012phase,
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@book{rovzdestvenskǐi1983systems,
 address = {Providence, RI},
 author = {B. L. Ro{\v z}destvenskǐi and N. N. Janenko},
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 title = {{S}ystems of {Q}uasilinear {E}quations and {T}heir {A}pplications to {G}as {D}ynamics},
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@article{roweis2000nonlinear,
 abstract = {Many areas of science depend on exploratory data analysis and visualization.
The need to analyze large amounts of multivariate data raises the
fundamental problem of dimensionality reduction: how to discover
compact representations of high-dimensional data. Here, we introduce
locally linear embedding (LLE), an unsupervised learning algorithm
that computes low-dimensional, neighborhood-preserving embeddings
of high-dimensional inputs. Unlike clustering methods for local dimensionality
reduction, LLE maps its inputs into a single global coordinate
system of lower dimensionality, and its optimizations do not involve
local minima. By exploiting the local symmetries of linear reconstructions,
LLE is able to learn the global structure of nonlinear manifolds,
such as those generated by images of faces or documents of text.},
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@article{rowley2003reduction,
 abstract = {We present a general method for analysing and numerically solving
partial differential equations with self-similar solutions. The method
employs ideas from symmetry reduction in geometric mechanics, and
involves separating the dynamics on the shape space (which determines
the overall shape of the solution) from those on the group space
(which determines the size and scale of the solution). The method
is computationally tractable as well, allowing one to compute self-similar
solutions by evolving a dynamical system to a steady state, in a
scaled reference frame where the self-similarity has been factored
out. More generally, bifurcation techniques can be used to find self-similar
solutions, and determine their behaviour as parameters in the equations
are varied. The method is given for an arbitrary Lie group, providing
equations for the dynamics on the reduced space, for reconstructing
the full dynamics and for determining the resulting scaling laws
for self-similar solutions. We illustrate the technique with a numerical
example, computing self-similar solutions of the Burgers equation},
 author = {Rowley, C. W. and Kevrekidis, I. G. and Marsden, J. E. and Lust, K.},
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 pages = {1257--1275},
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 year = {2003}
}

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poles (or resonances) are, however, not easy to locate. This note reviews
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@misc{saul2019scikittda,
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@conference{suri2015experimental,
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 url = {http://meetings.aps.org/link/BAPS.2015.DFD.D29.2},
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 abstract = {This is a statistical extension of the PIM method. Here,
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 title = {{S}uppression of period doubling in symmetric systems},
 volume = {52},
 year = {1984}
}

@article{swinney1983observations,
 author = {H. L. Swinney},
 journal = {Physica D},
 pages = {3--15},
 title = {{O}bservations of order and chaos in nonlinear systems},
 volume = {7},
 year = {1983}
}

@article{sylvester1852demonstration,
 author = {Sylvester, J. J.},
 journal = {Philos. Mag.},
 pages = {138--142},
 title = {{A} demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares},
 volume = {4},
 year = {1852}
}

@book{szebehely1967theory,
 address = {New York},
 author = {V. Szebehely},
 publisher = {Academic Press},
 title = {{T}heory of {O}rbits},
 year = {1967}
}

@article{tadrist2020predictability,
 author = {L. Tadrist and T. Gilet and P. Schlagheck and J. W. M. Bush},
 doi = {10.1103/PhysRevE.102.013104},
 issue = {1},
 journal = {Phys. Rev. E},
 numpages = {14},
 pages = {013104},
 title = {{P}redictability in a hydrodynamic pilot-wave system: {R}esolution of walker tunneling},
 volume = {102},
 year = {2020}
}

@article{tajima2002microextensive,
 author = {S. Tajima and H. S. Greenside},
 journal = {Phys. Rev. E},
 pages = {017205},
 title = {{M}icroextensive chaos of a spatially extended system},
 volume = {66},
 year = {2002}
}

@article{takahashi1984towards,
 author = {Takahashi, Y. and Oono, Y.},
 doi = {10.1143/PTP.71.851},
 journal = {Prog. Theor. Phys},
 pages = {851--854},
 title = {{T}owards the statistical mechanics of chaos},
 volume = {71},
 year = {1984}
}

@article{takavc1992invariant,
 author = {P. Tak{\'a}{\v c}},
 journal = {Nonlinearity},
 pages = {289--321},
 title = {{I}nvariant 2-tori in the time-dependent {G}inzburg-{L}andau equation},
 volume = {5},
 year = {1992}
}

@article{takeuchi2009lyapunov,
 abstract = {We show, using generic globally-coupled systems, that the collective
dynamics of large chaotic systems is encoded in their Lyapunov spectra:
most modes are typically localized on a few degrees of freedom, but
some are delocalized, acting collectively on the trajectory. For
globally-coupled maps, we show moreover a quantitative correspondence
between the collective modes and some of the so-called Perron-Frobenius
dynamics. Our results imply that the conventional definition of extensivity
must be changed as soon as collective dynamics sets in.},
 archiveprefix = {arXiv},
 author = {Takeuchi, K. A. and Ginelli, F. and Chat{\'e}, H.},
 eprint = {0907.4298},
 journal = {Phys. Rev. Lett.},
 pages = {154103},
 title = {{L}yapunov analysis captures the collective dynamics of large chaotic systems},
 volume = {103},
 year = {2009}
}

@article{takeuchi2011hyperbolic,
 addendum = {http://arXiv.org/abs/1107.2567},
 author = {Takeuchi, K. A. and Yang, H. L. and Ginelli, F. and Radons, G. and Chat{\'e}, H.},
 doi = {10.1103/PhysRevE.84.046214},
 journal = {Phys. Rev. E},
 pages = {046214},
 title = {{H}yperbolic decoupling of tangent space and effective dimension of dissipative systems},
 volume = {84},
 year = {2011}
}

@article{takhtajan1994on,
 archiveprefix = {arXiv},
 author = {Takhtajan, L.},
 eprint = {hep-th9301111},
 journal = {Commun. Math. Phys.},
 title = {{O}n foundation of the generalized {N}ambu mechanics},
 year = {1994}
}

@article{tambasco2016onset,
 author = {L. D. Tambasco and D. M. Harris and A. U. Oza and R. R. Rosales and J. W. M. Bush},
 doi = {10.1063/1.4964350},
 journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
 number = {10},
 pages = {103107},
 title = {{T}he onset of chaos in orbital pilot-wave dynamics},
 url = {https://doi.org/10.1063/1.4964350},
 volume = {26},
 year = {2016}
}

@book{taub1961john,
 address = {New York},
 editor = {A. H. Taub},
 publisher = {Pergamon Press},
 title = {{J}ohn von {N}eumann: {C}ollected {W}orks. {V}olume {I}I: {O}perators, {E}rgodic {T}heory and {A}lmost {P}eriodic {F}unctions in a {G}roup},
 year = {1961}
}

@article{taylor1923stability,
 author = {Taylor, G. I.},
 journal = {Phil. Trans. Royal Soc. A},
 pages = {289--343},
 title = {{S}tability of a viscous liquid contained between two rotating cylinders},
 volume = {223},
 year = {1923}
}

@article{taylor1938spectrum,
 author = {Taylor, G. I.},
 bdsk-url-1 = {http://dx.doi.org/10.1098/rspa.1938.0032},
 doi = {10.1098/rspa.1938.0032},
 journal = {Proc. R. Soc. Lond. A},
 pages = {476--490},
 title = {{T}he spectrum of turbulence},
 volume = {164},
 year = {1938}
}

@book{temam1988infinitedimensional,
 address = {New York},
 author = {R. Temam},
 publisher = {Springer},
 title = {{I}nfinite-{D}imensional {D}ynamical {S}ystems in {M}echanics and {P}hysics},
 year = {1988}
}

@article{temam1991approximation,
 abstract = {Estimates on the dimension of the attractors and the
decay rates of Fourier mode of NSe are given. By approximation of
the inertial manifold, the interaction between the large and small
eddies are derived. Such laws open the way to numerically efficient
algorithms via multiscale mthods.},
 author = {R. Temam},
 journal = {Proc. R. Soc. Lond. A},
 pages = {23--39},
 title = {{A}pproximation of attractors, large eddy simulations and multiscale methods},
 volume = {434},
 year = {1991}
}

@book{temme1996special,
 address = {New York},
 author = {N. M. Temme},
 publisher = {Wiley-Interscience},
 title = {{S}pecial functions: an introduction to the classical functions of mathematical physics},
 year = {1996}
}

@article{tempelmann2010spatial,
 author = {Tempelmann, D. and Hanifi, A. and Henningson, D. S.},
 bdsk-url-1 = {http://dx.doi.org/10.1017/S0022112009993260},
 doi = {10.1017/S0022112009993260},
 journal = {J. Fluid Mech.},
 pages = {5--37},
 title = {{S}patial optimal growth in three-dimensional boundary layers},
 volume = {646},
 year = {2010}
}

@article{tenenbaum2000global,
 abstract = {Scientists working with large volumes of high-dimensional data, such
as global climate patterns, stellar spectra, or human gene distributions,
regularly confront the problem of dimensionality reduction: finding
meaningful low-dimensional structures hidden in their high-dimensional
observations. The human brain confronts the same problem in everyday
perception, extracting from its high-dimensional sensory inputs -30,000
auditory nerve fibers or 106 optic nerve fibers- manageably small
number of perceptually relevant features. Here we describe an approach
to solving dimensionality reduction problems that uses easily measured
local metric information to learn the underlying global geometry
of a data set. Unlike classical techniques such as principal component
analysis (PCA) and multidimensional scaling (MDS), our approach is
capable of discovering the nonlinear degrees of freedom that underlie
complex natural observations, such as human handwriting or images
of a face under different viewing conditions. In contrast to previous
algorithms for nonlinear dimensionality reduction, ours efficiently
computes a globally optimal solution, and, for an important class
of data manifolds, is guaranteed to converge asymptotically to the
true structure.},
 author = {Tenenbaum, J. B. and Silva, V. de and Langford, J. C.},
 doi = {10.1126/science.290.5500.2319},
 journal = {Science},
 pages = {2319--2323},
 title = {{A} global geometric framework for nonlinear dimensionality reduction},
 volume = {290},
 year = {2000}
}

@book{tennekes1972first,
 addr = {Cambridge, MA},
 author = {H. Tennekes and J.L. Lumley},
 publisher = {MIT Press},
 title = {{A} First {C}ourse in {T}urbulence},
 year = {1972}
}

@article{thiffeault2001geometrical,
 abstract = {Constraints are found on the spatial variation of finite-time Lyapunov
exponents of two and three-dimensional systems of ordinary differential
equations. In a chaotic system, finite-time Lyapunov exponents describe
the average rate of separation, along characteristic directions,
of neighboring trajectories. The solution of the equations is a coordinate
transformation that takes initial conditions (the Lagrangian coordinates)
to the state of the system at a later time (the Eulerian coordinates).
This coordinate transformation naturally defines a metric tensor,
from which the Lyapunov exponents and characteristic directions are
obtained. By requiring that the Riemann curvature tensor vanish for
the metric tensor (a basic result of differential geometry in a flat
space), differential constraints relating the finite-time Lyapunov
exponents to the characteristic directions are derived. These constraints
are realized with exponential accuracy in time. A consequence of
the relations is that the finite-time Lyapunov exponents are locally
small in regions where the curvature of the stable manifold is large,
which has implications for the efficiency of chaotic mixing in the
advection-diffusion equation. The constraints also modify previous
estimates of the asymptotic growth rates of quantities in the dynamo
problem, such as the magnitude of the induced current.},
 archiveprefix = {arXiv},
 author = {Thiffeault, J.-L. and Boozer, A. H.},
 eprint = {physics/0009017},
 journal = {Chaos},
 pages = {16--28},
 title = {{G}eometrical constraints on finite-time {L}yapunov exponents in two and three dimensions},
 volume = {11},
 year = {2001}
}

@article{thiffeault2002derivatives,
 abstract = {In a smooth flow, the leading-order response of trajectories to infinitesimal
perturbations in their initial conditions is described by the finite-time
Lyapunov exponents and associated characteristic directions of stretching.
We give a description of the second-order response to perturbations
in terms of Lagrangian derivatives of the exponents and characteristic
directions. These derivatives are related to generalised Lyapunov
exponents, which describe deformations of phase space elements beyond
ellipsoidal. When the flow is chaotic, care must be taken in evaluating
the derivatives because of the exponential discrepancy in scale along
the different characteristic directions. Two matrix decomposition
methods are used to isolate the directions of stretching, the first
appropriate in finding the asymptotic behaviour of the derivatives
analytically, the second better suited to numerical evaluation. The
derivatives are shown to satisfy differential constraints that are
realised with exponential accuracy in time. With a suitable reinterpretation,
the results of the paper are shown to apply to the Eulerian framework
as well.},
 archiveprefix = {arXiv},
 author = {Thiffeault, J.-L. },
 eprint = {nlin/0101012},
 journal = {Physica D},
 pages = {139--161},
 title = {{D}erivatives and constraints in chaotic flows: asymptotic behaviour and a numerical method},
 volume = {172},
 year = {2002}
}

@article{thiffeault2012using,
 archiveprefix = {arXiv},
 author = {Thiffeault, J.-L.},
 bdsk-url-1 = {http://dx.doi.org/10.1088/0951-7715/25/2/R1},
 doi = {10.1088/0951-7715/25/2/R1},
 eprint = {1105.1101},
 journal = {Nonlinearity},
 pages = {1--44},
 title = {{U}sing multiscale norms to quantify mixing and transport},
 volume = {25},
 year = {2012}
}

@book{thomas1995numerical,
 address = {New York},
 author = {J. W. Thomas},
 publisher = {Spring-Verlag},
 title = {{N}umerical {P}artial {D}ifferential {E}quations},
 year = {1995}
}

@article{thomson1887rectilineal,
 author = {W. Thomson},
 journal = {Philos. Mag.},
 pages = {188--196},
 title = {{R}ectilineal {M}otion of {V}iscous {F}luid between two {P}arallel {P}lanes},
 volume = {24},
 year = {1887}
}

@article{thoo1998picture,
 author = {Thoo, J. B.},
 journal = {College Math. J.},
 pages = {408--411},
 title = {{A} picture is worth a thousand words},
 volume = {29},
 year = {1998}
}

@article{tillmark1992experiments,
 author = {N. Tillmark and P. H. Alfredsson},
 journal = {J. Fluid Mech.},
 pages = {89--102},
 title = {{E}xperiments on transition in plane {C}ouette flow},
 volume = {235},
 year = {1992}
}

@article{tillmark1995on,
 author = {N. Tillmark},
 journal = {Europhys. Lett.},
 pages = {481--485},
 title = {{O}n the spreading mechanisms of a turbulent spot in plane {C}ouette flow},
 volume = {32},
 year = {1995}
}

@book{tinkham2003group,
 address = {New York},
 author = {Tinkham, M.},
 publisher = {Dover},
 title = {{G}roup {T}heory and {Q}uantum {M}echanics},
 year = {2003}
}

@article{toh2003periodiclike,
 author = {S. Toh and T. Itano},
 journal = {J. Fluid Mech.},
 pages = {67--76},
 title = {{A} periodic-like solution in channel flow},
 volume = {481},
 year = {2003}
}

@article{tompaidis1996numerical,
 abstract = {Newton method is used to construct periodic orbits of
longer and longer period to approach a invariant torus with specific
rotation vector. Behavior after the break of a torus is described.},
 author = {S. Tompaidis},
 journal = {Experimental Mathematics},
 pages = {211--230},
 title = {{N}umerical {S}tudy of {I}nvariant {S}ets of a {Q}uasi-periodic {P}erturbation of a {S}ymplectic {M}ap},
 volume = {5},
 year = {1996}
}

@article{topaz2015topological,
 archiveprefix = {arXiv},
 author = {C. M. Topaz AND L. Ziegelmeier AND T. Halverson},
 doi = {10.1371/journal.pone.0126383},
 eprint = {1412.6430},
 journal = {PLOS ONE},
 month = {05},
 number = {5},
 pages = {1--26},
 publisher = {Public Library of Science},
 title = {{T}opological {D}ata {A}nalysis of {B}iological {A}ggregation {M}odels},
 url = {https://doi.org/10.1371/journal.pone.0126383},
 volume = {10},
 year = {2015}
}

@article{torcini1996order,
 abstract = {the maximal consersed phase gradient is introduced as an
order parameter to characterize the transition from phase to defect
turbulence in the cgle. it has a finite value in the pt regime and
decreases to zero when the transition to defect turbulence is
approached. a modified kse is able to reproduce the main feature of
the stable waves and to explain their origin.},
 author = {A. Torcini},
 journal = {Phys. Rev. Lett.},
 pages = {1047},
 title = {{O}rder parameter for the transition from phase to amplitude turbulence},
 volume = {77},
 year = {1996}
}

@article{torciniundfrauenkron1997studies,
 abstract = {The phase gradient is used as an order parameter.
Different states are identified. In the PT region, a modified KSe
rules the phase dynamics of the CGLe.},
 author = {A. Torcini und H. Frauenkron and P. Grassberger},
 journal = {Phys. Rev. E},
 pages = {5073},
 title = {{S}tudies of phase turbulence in the one dimensional complex {G}inzburg-{L}andau equation},
 volume = {55},
 year = {1997}
}

@article{toronov1994geometric,
 abstract = {Pancharatnam\textquotesingle          s geometric phase is introduced for such nonlinear dissipative
systems as lasers and liquid flows. Two types of geometric; phases
are shown to arise in these systems: the phase induced by the inner
dynamics of the system and the one caused by the cyclic and adiabatic
variation of the system parameters. A possible generalization of
the geometric-effects theory in other dissipative systems is discussed.},
 author = {Toronov, V. Y. and Derbov, V. L.},
 journal = {Phys. Rev. E},
 pages = {1392--1399},
 title = {{G}eometric phases in lasers and liquid flows},
 volume = {49},
 year = {1994}
}

@article{toronov1994geometricphase,
 abstract = {We show that such phenomena of laser dynamics as mean-phase-slope
jumps and temporal phase jumps at resonance between the cavity and
spectral line frequencies are intrinsically connected with the topology
of attractors in the space of rays and can be interpreted as the
manifestations of the geometric-phase properties of the evolution
operator.},
 author = {Toronov, V. Y. and Derbov, V. L.},
 journal = {Phys. Rev. A},
 keywords = {complex Lorenz, geometric phase, Laser, symmetry},
 pages = {878--881},
 title = {{G}eometric-phase effects in laser dynamics},
 volume = {50},
 year = {1994}
}

@article{toronov1997boundedness,
 author = {Toronov, V. Y. and Derbov, V. L.},
 journal = {Phys. Rev. E},
 pages = {3689--3692},
 title = {{B}oundedness of attractors in the complex {L}orenz model},
 volume = {55},
 year = {1997}
}

@article{toronov1997geometric,
 abstract = {An investigation is made of the geometric phases in a ring laser with
counterpropagating waves. It is shown that the frequency splitting
of the counterpropagating waves that appears in the case of radiation
pulsations or is induced by displacement of an external mirror can
be regarded as a manifestation of a geometric phase.},
 author = {Toronov, V. Y. and Derbov, V. L.},
 journal = {Quantum Electronics},
 pages = {644--648},
 title = {{G}eometric phases in a ring laser},
 volume = {27},
 year = {1997}
}

@article{toronov1998topological,
 author = {Toronov, V. Y. and Derbov, V. L.},
 journal = {J. Optical Soc. of America B},
 pages = {1282--1290},
 title = {{T}opological properties of laser phase},
 volume = {15},
 year = {1998}
}

@article{townsend1961equilibrium,
 author = {A. A. Townsend},
 doi = {10.1017/S0022112061000883},
 journal = {J. Fluid Mech.},
 pages = {97-120},
 title = {{E}quilibrium layers and wall turbulence},
 volume = {11},
 year = {1961}
}

@book{townsend1976structure,
 address = {Cambridge},
 author = {A. A. Townsend},
 publisher = {Cambridge Univ. Press},
 title = {{T}he {S}tructure of {T}urbulent {S}hear {F}lows, 2\textbackslash          textsuperscriptnd {E}dn.},
 year = {1976}
}

@unpublished{tranchida2014knotted,
 archiveprefix = {arXiv},
 author = {Tranchida, J. and Thibaudeau, P. and Nicolis, S.},
 eprint = {1404.7774},
 title = {{K}notted strange attractors and matrix {L}orenz systems},
 year = {2014}
}

@article{trefethen1993hydrodynamic,
 author = {N. Trefethen and A. E. Trefethen and S. C. Reddy and T. A. Driscoll},
 journal = {Science},
 pages = {578--584},
 title = {{H}ydrodynamic stability without eigenvalues},
 volume = {261},
 year = {1993}
}

@book{trefethen1997numerical,
 addr = {Philadelphia},
 author = {L. N. Trefethen and D. Bau},
 publisher = {SIAM},
 title = {{N}umerical {L}inear {A}lgebra},
 year = {1997}
}

@article{trefethen1997pseudospectra,
 author = {L. Trefethen},
 journal = {SIAM Rev.},
 pages = {383--406},
 title = {{P}seudospectra of linear operators},
 volume = {39},
 year = {1997}
}

@book{trefethen2000spectral,
 address = {Philadelphia},
 author = {L. N. Trefethen},
 publisher = {SIAM},
 title = {{S}pectral methods in {M}atlab},
 year = {2000}
}

@book{trefethen2005spectra,
 address = {Princeton},
 author = {L. N. Trefethen and M. Embree},
 publisher = {Princeton Univ. Press},
 title = {{S}pectra and pseudospectra: the behavior of nonnormal matrices and operators},
 year = {2005}
}

@article{troy1989existence,
 abstract = {For c=1, the existence of two periodic orbits and two
heteroclinic orbits are proved and suggestions for
proving the existence of more complicated orbits are
given.},
 author = {W. C. Troy},
 doi = {10.1016/0022-0396(89)90134-4},
 journal = {J. Diff. Eqn.},
 pages = {269--313},
 title = {{T}he existence of steady solutions of the {K}uramoto-{S}ivashinsky equation},
 volume = {82},
 year = {1989}
}

@article{tu2014on,
 archiveprefix = {arXiv},
 author = {J. H. Tu and C. W. Rowley and D. M. Luchtenburg and S. L. Brunton and J. N. Kutz},
 eprint = {1312.0041},
 journal = {J. Comput. Dyn.},
 number = {2},
 pages = {391--421},
 title = {{O}n dynamic mode decomposition: {T}heory and applications},
 volume = {1},
 year = {2014}
}

@article{tucker2002rigorous,
 address = {New York},
 author = {Tucker, W.},
 journal = {Found. Comp. Math},
 pages = {53--117},
 publisher = {Springer},
 title = {{A} rigorous {O}DE solver and {S}male\textquotesingle          s 14th problem},
 volume = {2},
 year = {2002}
}

@article{tuckerman2002symmetry,
 archiveprefix = {arXiv},
 author = {L. S. Tuckerman and D. Barkley},
 eprint = {physics/0312051},
 journal = {Theor. Comp. Fluid Dyn.},
 pages = {91--97},
 title = {{S}ymmetry breaking and turbulence in perturbed plane {C}ouette flow},
 volume = {16},
 year = {2002}
}

@article{tuckerman2007mean,
 author = {L. S. Tuckerman and D. Barkley},
 journal = {Springer Proc. Physics},
 pages = {224--226},
 title = {{M}ean flow and modeling of turbulent-laminar patterns in plane {C}ouette flow},
 volume = {117},
 year = {2007}
}

@article{tuckerman2014turbulentlaminar,
 author = {Tuckerman, L. S. and Kreilos, T. and Schrobsdorff, H. and Schneider, T. M. and Gibson, J. F.},
 doi = {http://dx.doi.org/10.1063/1.4900874},
 journal = {Phys. Fluids},
 pages = {114103},
 title = {{T}urbulent-laminar patterns in plane {P}oiseuille flow},
 volume = {26},
 year = {2014}
}

@book{tung1985group,
 address = {Singapore},
 author = {Tung, W.-K.},
 publisher = {World Scientific},
 title = {{G}roup {T}heory in {P}hysics},
 year = {1985}
}

@book{vallero2008fundamentals,
 address = {New York},
 author = {Vallero, D.},
 edition = {4\textbackslash          textsuperscriptth},
 publisher = {Academic Press},
 title = {{F}undamentals of {A}ir {P}ollution},
 year = {2008}
}

@phdthesis{vandoorne2004stereoscopic,
 author = {C. W. H. van Doorne},
 school = {Delft},
 title = {{S}tereoscopic {P}IV on transition in pipe flow},
 url = {www.ahd.tudelft.nl},
 year = {2004}
}

@incollection{vanhecke1994amplitude,
 address = {Amsterdam},
 author = {M. van Hecke and P. C. Hohenberg and W. van Saarloos},
 booktitle = {Fundamental Problems in Statistical Mechanics},
 editor = {H. van Beijeren and M. H. Ernst},
 publisher = {North-Holland},
 title = {{A}mplitude equations for pattern forming systems},
 volume = {VIII},
 year = {1994}
}

@article{vanhecke1998building,
 author = {M. van Hecke},
 journal = {Phys. Rev. Lett.},
 pages = {1896},
 title = {{B}uilding blocks of spatiotemporal intermittency},
 volume = {80},
 year = {1998}
}

@article{vanhecke2001ordered,
 abtract = {Ordered hole-defect dynamics happens when an unstable hole invade a plane wave state and periodically nucleates defects from which new holes are born. The interaction between the holes and a self-disordered background are essential for the occurrence of spatiotemporal chaos in hole-defect states.},
 author = {M. van Hecke and M. Howard},
 journal = {Phys. Rev. Lett.},
 pages = {2018},
 title = {{O}rdered and self-disordered dynamics of holes and defects in the one-dimensional complex {G}inzburg-{L}andau equation},
 volume = {86},
 year = {2001}
}

@article{vanhecke2003coherent,
 abstract = {Give an overview of the most important properties of the
coherent structures and address the question how such structures may
be identified in the experimental space-time data set.},
 author = {M. van Hecke},
 journal = {Physica D},
 pages = {134},
 title = {{C}oherent and incoherent structures in systems described by the 1{D} C{G}LE: {E}xperiments and identification},
 volume = {174},
 year = {2003}
}

@article{vansaarloos1990pulses,
 author = {W. van Saarloos and P. C. Hohenberg},
 journal = {Phys. Rev. Lett.},
 pages = {749},
 title = {{P}ulses and {F}ronts in the {C}omplex {G}inzburg-{L}andau {E}quation near a {S}ubcritical {B}ifurcation},
 volume = {64},
 year = {1990}
}

@article{vansaarloos1992fronts,
 abstract = {Propose the concept of coherent structures and study the
competition between them in chaotic dynamics. Specifically, the
invasion of front to another state is discussed.},
 author = {W. van Saarloos and P. C. Hohenberg},
 journal = {Physica D},
 pages = {303--367},
 title = {{F}ronts, pulses, sources and sinks in generalized complex {G}inzburg-{L}andau equations},
 volume = {56},
 year = {1992}
}

@article{vanveen2011homoclinic,
 archiveprefix = {arXiv},
 author = {L. van Veen and G. Kawahara},
 doi = {10.1103/PhysRevLett.107.114501},
 eprint = {1103.1331},
 journal = {Phys. Rev. Lett.},
 pages = {114501},
 primaryclass = {physics.flu-dyn},
 title = {{H}omoclinic tangle on the edge of shear turbulence},
 volume = {107},
 year = {2011}
}

@article{vanveen2011homoclinicb,
 author = {L. van Veen and G. Kawahara},
 journal = {Phys. Rev. Lett.},
 pages = {114501},
 title = {{H}omoclinic tangle on the edge of shear turbulence},
 volume = {107},
 year = {2011}
}

@article{vanveen2011on,
 author = {L. van Veen and G. Kawahara and M. Atsushi},
 doi = {10.1137/100789804},
 journal = {SIAM J. Sci. Comput.},
 pages = {25--44},
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@article{viswanath2002lindstedtpoincare,
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@article{viswanath2003symbolic,
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@article{viswanath2004fractal,
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@article{wittenberg1999scale,
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In large scales, the dynamics is Gaussian.
In the intermediate scales, the dynamics is reminiscent of travelling waves
and heteroclinic cyces which is the typical behavior for small system size.
In the small scales, the dynamics is intermittent. Through investigation of
the interaction between different scales, we see the intermediate structures
give the defining shape of the cell and the large scales trigger
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 author = {R. W. Wittenberg and P. Holmes},
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Standard techniques for computing Lyapunov vectors produce
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asymptotic singular vectors is proposed. The Lyapunov
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defined, in principle, for all time through the tangent
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construction of the n most rapidly decaying Lyapunov
vectors from n decaying forward and n-1 decaying backward
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