From c4152381d755ca400c7a73799a7e4484d6810bf4 Mon Sep 17 00:00:00 2001 From: pclus Date: Fri, 27 Sep 2024 10:25:39 +0200 Subject: [PATCH] Initial commit --- Manifest.toml | 2115 +++++++++++++++++++++++++ Project.toml | 14 + README.md | 90 ++ src/.KuramotoPopulationNetwork.jl.swp | Bin 0 -> 16384 bytes src/KuramotoPopulationNetwork.jl | 286 ++++ 5 files changed, 2505 insertions(+) create mode 100644 Manifest.toml create mode 100644 Project.toml create mode 100644 README.md create mode 100644 src/.KuramotoPopulationNetwork.jl.swp create mode 100644 src/KuramotoPopulationNetwork.jl diff --git a/Manifest.toml b/Manifest.toml new file mode 100644 index 0000000..72fcda9 --- /dev/null +++ b/Manifest.toml @@ -0,0 +1,2115 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "7b4c2a823ff7193ea4b1ae903a5de431ef39a6a4" + +[[deps.ADTypes]] +git-tree-sha1 = "eea5d80188827b35333801ef97a40c2ed653b081" +uuid = 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b/README.md @@ -0,0 +1,90 @@ +# Julia code to simulate networks of Kuramoto-Sakaguchi populations + + +This repository contains the code accompaining the paper [From chimeras to extensive chaos in networks of heterogeneous +Kuramoto oscillator populations](https://arxiv.org/abs/2407.20408) by Pol Floriach, Jordi Garcia-Ojalvo, and Pau Clusella. + +Two different set of codes are provided: + +1. Some `auto-07p` files and instructions to obtain the main bifurcations displayied in the paper for a two-population model (Fig. 2). These is all provided in the `auto` folder. See `diagram.md` in that folder. +2. The Julia code to reproduce most of the simulations in the paper. These functions are provided in the `KuramotoPopulationNetwork.jl` module in the `src` directory. In the following we show how to use the functions in that module to simulate the system using some examples. + + +## Using the `KuramotoPopulationNetwork` module + +Open a Julia REPL in the `src` folder and then use the following +to include it in the path and load it, alongside other packages. +Notice that you might have to install new packages (see the `KuramotoPopulationNetwork.jl` file to identify which ones). + +```julia +push!(LOAD_PATH, pwd()) +using KuramotoPopulationNetwork +using Graphs +using Plots,ColorSchemes +``` + +Then we can use `Graphs.jl` to generate a ring network with nearest neighbours: + +```julia +N = 128; +g_ring = watts_strogatz(N, 2, 0.0); +``` + +Now we can reproduce of Fig. 5(a) in the paper: + +```julia +t,R,ϕ,u0 = simulation(;K=7.0,p=0.9,α=1.2,gr=g_ring,trans=5e2,tmax=4e2,ic="homogeneous"); +n = size(R,1) +range = n-10000:1:n; +heatmap(1:128,t[range],ϕ[range,:],c=:cyclic_mrybm_35_75_c68_n256) +heatmap(1:128,t[range],R[range,:],c=:linear_bmy_10_95_c78_n256) +``` + +We can compute and plot the full Lyapunov spectra for this case (this can take some minutes): + +```julia +λ, = simulationLE(;K=7,p=0.9,α=1.2,gr=g_ring,trans=2e2,tmax=1e3,nLE=2*size(g_ring,1),ic=u0); +plot(λ) +``` +With this data we can compute the attractor dimension and the dynamical entropy: + +```julia +dimKY(λ) +entropy(λ) +``` + +Reproduction of Fig. 5(b) in the paper: +```julia +@time t,R,ϕ,u0 = simulation(;K=15.0,p=0.5,α=1.2,gr=g_ring,trans=5e2,tmax=4e2,ic="homogeneous"); +n = size(R,1) +range = n-10000:1:n; +heatmap(1:128,t[range],ϕ[range,:],c=:cyclic_mrybm_35_75_c68_n256) +heatmap(1:128,t[range],R[range,:],c=:linear_bmy_10_95_c78_n256) +``` + +Simulation on the slow regime that appears for p=0.5. Notice we change `tmax` and `dt`: +```julia +t,R,ϕ,u0 = simulation(;K=20.0,p=0.5,α=1.2,gr=graph,trans=0e3,tmax=40e3,dt=1e-1,ic="homogeneous"); +n = size(R,1) +range = n-1000:10:n; +heatmap(1:128,t[range],ϕ[range,:],c=:cyclic_mrybm_35_75_c68_n256) +heatmap(1:128,t[range],R[range,:],c=:linear_bmy_10_95_c78_n256) +``` + +Let's compute now the LE for a simulation of the ER network: +```julia +n = 128; +avk = 10; +g_er = erdos_renyi(n, avk*1.0/n); + +t,R,ϕ,u0 = simulation(;K=14.0,p=0.9,α=1.2,gr=g_er,trans=5e2,tmax=4e2,ic="homogeneous"); +λ, = simulationLE(;K=14,p=0.9,α=1.2,gr=g_er,trans=2e2,tmax=1e3,nLE=2*size(g_er,1),ic=u0); +``` +and plot the results: +```julia +n = size(R,1) +trange = n-10000:1:n; +heatmap(1:128,t[trange],ϕ[trange,:],c=:cyclic_mrybm_35_75_c68_n256) +heatmap(1:128,t[trange],R[trange,:],c=:linear_bmy_10_95_c78_n256) +plot(λ) +``` diff --git a/src/.KuramotoPopulationNetwork.jl.swp b/src/.KuramotoPopulationNetwork.jl.swp new file mode 100644 index 0000000000000000000000000000000000000000..d7fb5ffcd77e05a5a638486ef5f34fc78b73a86d GIT binary patch literal 16384 zcmeHOZ>$_u6(1~!px_^j0*T3?FSfJ$X7}xG0qbiA`ubim?Q0+9wIZ-Qy7P8scV|00 zv&_uC?luh)(GN8mKfEH!7ZZqpF~Kw#kra$y{1c7aV3e3B67h@1^n)K%{1fnZ?wy(a z)Aw30Cgxv@6v>s)0lUS1H>{#R7kw*n9(B^Px{CJ?TxVfm8#j22u^A z8b~#eY9Q4>s)1Ak|Nj~Y+nbcnVw&%Yrh0eudCA!48_^Z_-!s<#dek2u#-ADMkG`fK zsRmLFq#8&ykZK^+K&pXM1E~g54Wt@KHIQl`)xiHi1BR(6S7F>Q^MN0~|HtS1SKhBE zzXzTLP6GD=W#CHS3gD%y73G(}v%vR&F94qhrhy-U;pc$cfPcSFQN9a&2KdK&73FKd zG;j&<@|BA6JK#6KbHH)nPGB2wHSp&v6y?Xj;$d@t_6PnZbexIssI81xlB=B1AYZO44eYmzyV-C@Ckr`%Yi>! zswiiGQ-BM!fR6%~0Z(tjJOKsx778j)0bd1900)4N0q3CS7Xa4xmssb+7Y(jmw!7?g zTr;2pEKu&AZ=ZZ1??1kxb#WDW7-xU3T|IE7dkQOo* z@;7DYv$ko~L6}vGxprnCe$DgACfg;)4#IL%PK+Cy+*q7mAS?i`T{R;JTSnNRU3oIW zHP>IINh;+FSyiQ8Hn83P>0CEk^@9HCOg8AEQ_c0{ceIOswD&UTn9D-4^VwFxmd>6` z*v5~=Y+|+)4TJU(wzF%qU1n*A*g{$%<>WvrA?+Ho5Yp~mo2@aXgn06k1F57EqG5(+ zA=u6&Y=@VrEz~kW%Z^t)N~UepJ=btF;wACgq202Nix+gLi#cf2g#t9{;hyj&5cTkt zs$R!2Y1v|9t$M9?C$!9H5fjVi6E6HOSICU%I%+eiz#TF}#GwB6dQHm4ob?$8#%t!?;9t>aet z{VCdKRJ~=}aF5(R<67YjnA3^|0xiran>TM3vJkE;Rl*9{0fh~Vu(}ql7`|;RJ63*} z22WPVz;0$Td@)*9X!!woHPI6Eg3xMdWCXlvD{{z6o`zYhB*Uy&)sU^JMnPzVgAs#~ zArA<8gRtj)yGzF!Ry2G(GnP64Bg@PyGqBNKXjcs*o45W9HippSs0ZxPq)xNa&&|QDtwDm_uhWbq z*}VoyVy!{1Jk7yEI_k3U(22heIxV$V7T(!jTFOkevpu+Rp-~AUV6X=H?jdrz zIQh(oFoOH!1%$bS$xGR8c8&Eq$bD)z+vWMFy=*TdEpcQvCVLjdsB9NR{nfSjnCw{) z9R&#}qoI;D5iF#0GS~G^f=NOY`zN^=38xKvIl}c$IJwBTnJXIMAGL~V1C324y`6Y7awwuF) zzGbm1g`>@M1NX(JQ=(8~qgPCBXZ*nKk%} ztpn(`Cx?uqM!i#Q*wk(qfki>Xu7y`=|O?(c|sL`=P)YU6!o1dO3Lld$QpCTtoU+Pu2uT6Wlr6(A!Vsr^$N z;bauVqOCREcESlSP}g!n=v!_bd8^ug{-V#$C(qU$ZyD1-F4cByw~j~`v5BT51sZ4@ zb}oW6*505H5AjMFAsD!!VY>*G){4c78H(P#B11+Q1%6@do^QhZLyzoM+xJ!w^amC? 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z5A7lDkNYq|VPXrJ$q_~aRV36ZnL9-$DKd=Uem_*rXrsj=@-HjJTq)aMol@udhb9%j zhsmwLBqIx^#1HYOoC?nZlgO5`XCGWcfT(*1%#&~7w*9Tr`7w)FS+YxN3TcrRZlEKI{}(M^qZa@G literal 0 HcmV?d00001 diff --git a/src/KuramotoPopulationNetwork.jl b/src/KuramotoPopulationNetwork.jl new file mode 100644 index 0000000..6898f87 --- /dev/null +++ b/src/KuramotoPopulationNetwork.jl @@ -0,0 +1,286 @@ +module KuramotoPopulationNetwork + +using DifferentialEquations, Kronecker +using Graphs, LinearAlgebra, DynamicalSystems +using DelimitedFiles,SparseArrays,Statistics + +export OAnet!, Jacobian_OAnet!, simulation, simulationLE, statsR, dimKY, entropy + +# ------------------------------------------------ +""" + OAnet!(dv,v,par,t) + +Coupled Ott-Antonsen equations for integration. + + * `dv` and `v` are the velocity and state variables respectively. + * `par` is a tuple that contains several parameters and auxiliary variables in order to improve performance: + `K` is the global coupling strength of the system. + `p` is the self-strength (μ in the paper). + `Γ` is the heterogeneity of the system (Δ in the paper). + `ω` is the mean natural frequency of the oscillators. + `cosα` and `sinα`are the cosine and sine of the Sakaguchi phase shift `α` + `list` is the adjacency list of the graph (see the documentation of Graphs.jl). + `ideg` is a vector `ideg[i]=1/k[i]` where `k[i]` is the in-degree of node `i`. + `sx,sy,A,B,C,D,P,Q` are N-dimensional auxiliary vectors used for performance. + `N` is the number of nodes in the graph. + * `t` is the time variable. + +""" +function OAnet!(dv, v, par, t) + K,p,Γ,ω,cosα,sinα, list,ideg,sx,sy,A,B,C,D,P,Q,N = par + @inbounds x = @view v[1:2:end] + @inbounds y = @view v[2:2:end] + @inbounds dx = @view dv[1:2:end] + @inbounds dy = @view dv[2:2:end] + @fastmath @. dx = -Γ*x - ω*y + @fastmath @. dy = ω*x - Γ*y + + @fastmath @. P = (x*x) - (y*y) + @fastmath @. Q = 2*x*y + @fastmath @. A = cosα*( 1-P ) + Q*sinα + @fastmath @. B = sinα*( 1-P ) - Q*cosα + @fastmath @. C = sinα*(-1-P ) - Q*cosα + @fastmath @. D = cosα*( 1+P ) - Q*sinα + + @fastmath @inbounds sx[:] = 0.5*K*[(1-p)*ideg[i]*sum(x[list[i]])+p*x[i] for i in 1:N] + @fastmath @inbounds sy[:] = 0.5*K*[(1-p)*ideg[i]*sum(y[list[i]])+p*y[i] for i in 1:N] + @fastmath @. dx += A*sx + B*sy + @fastmath @. dy += C*sx + D*sy +end +# ------------------------------------------------ + +# ------------------------------------------------ +""" + Jacobian_OAnet!(M,v,par,t) + +Analytical Jacobian of the system for the computation of Lyapunov exponents. + * `M` is a matrix where the Jacobian is stored. + * `v` is the vector of state variables. + * `par` are the parameters of the system, as in `OAnet!`. + * `t` is the time variable. +""" +function Jacobian_OAnet!(M,v,par,t) + K,p,Γ,ω,cosα,sinα, list,ideg,sx,sy,A,B,C,D,P,Q,N = par + + @inbounds x = @view v[1:2:end] + @inbounds y = @view v[2:2:end] + + d0 = diagind(M); + d1 = diagind(M,1); + ds1 = diagind(M,-1); + @inbounds a = view(M,d0[1:2:end]) + @inbounds b = view(M,d1[1:2:end]) + @inbounds c = view(M,ds1[1:2:end]) + @inbounds d = view(M,d0[2:2:end]) + + # block diagonal, for optimization xcos, xsin, ycos and ysin could be precomputed + pre = 0.5*K*p + @fastmath @. a = -Γ + 2.0*((-x*cosα+y*sinα)*sx + (-x*sinα-y*cosα)*sy) + A*pre + @fastmath @. b = -ω + 2.0*(( y*cosα+x*sinα)*sx + ( y*sinα-x*cosα)*sy) + B*pre + @fastmath @. c = ω + 2.0*((-x*sinα-y*cosα)*sx + ( x*cosα-y*sinα)*sy) + C*pre + @fastmath @. d = -Γ + 2.0*(( y*sinα-x*cosα)*sx + (-y*cosα-x*sinα)*sy) + D*pre + + for j in 1:N + for i in list[j] + @inbounds pre = 0.5*K*ideg[i]*(1-p) + @inbounds M[2*i-1, 2*j-1] = pre*A[i] + @inbounds M[2*i-1, 2*j ] = pre*B[i] + @inbounds M[2*i , 2*j-1] = pre*C[i] + @inbounds M[2*i , 2*j ] = pre*D[i] + end + end +end +# ------------------------------------------------ + +# ------------------------------------------------ +""" + simulation(;kwargs...) + +Performs a simulations of the system with the given parameters and network topology. +The (optional) function arguments are: + * `K` the coupling of the system. + * `p` the self-coupling strength (μ in the paper). + * 'α' the Sakaguchi phase shift. + * 'gr' a Graph object containing the network topology, see the documentation of Graphs.jl. The row-normalization of the inputs in the connectivity matrix is computed authomatically, thus the connectivity of `gr` does not need to be row-normalized. + * 'tmax' simulation time after transient. + * 'trans' transient simulation time. + * 'ic' initial condition of the system. Can be: + - A vector containing the state variables to initialize the system. + - The String "random" for totally random initial values. + - The String "antiphase" for simulations starting close to the antiphase state. + - Any other string, which will cause the system to initialize close to the homogeneous state. + * `dt` integration time step. + + +Returns: + * A time vector. + * A matrix `R` with the values of the Kuramoto order parameters at all times and nodes. + * A matrix `Φ` with the values of the collective phase at all times and nodes. + * The final state of the system `u0`. +""" +function simulation(;K=20.0,p=0.9,α=1.45,gr=erdos_renyi(20, 0.2),tmax=200.0,trans=0.0,ic="random",dt=0.01) + N = length(gr.fadjlist) + Γ = 1.0 + ω = tan(α)*(K*cos(α)-Γ) + deg = degree(gr); + ideg = 1.0./deg; + par = K,p,Γ,ω,cos(α),sin(α),gr.fadjlist,ideg,zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),N; + + if typeof(ic)==String || length(ic)!=2*N + if ic=="random" + r0 = rand(N); # random r's + ϕ0 = rand(N)*2*π; + else + d = 1-2*Γ/(K*cos(α)) + r0 = ( d<0 ? 0.0 : sqrt(d)) + r0 = @. r0*ones(N)+1e-3*randn(N); + r0 = @. abs(r0); + ϕ0 =1e-3*randn(N) # random perturbation + if ic=="antiphase" + @. ϕ0[1:2:N]=ϕ0[1:2:N]+π # random perturbation + end + end + x0 = @. r0*cos(ϕ0); + y0 = @. r0*sin(ϕ0); + + u0 = [ (i%2==1 ? x0[Int(ceil(i/2))] : y0[Int(i/2)] ) for i in 1:2*N]; + else + u0 = ic + end + + + ds=ContinuousDynamicalSystem(OAnet!, u0, par; + diffeq=(alg=RK4(),dt=dt,adaptive=false)) + + ddt=(dt < 0.01 ? 0.01 : dt); # sample step (equivalent to "saveat") + tr=trajectory(ds, tmax ;Δt=ddt,Ttr=trans) + R = [ @. sqrt(tr[1][:,2*i-1]^2+tr[1][:,2*i]^2) for i in 1:N ]; + ϕ = [ @. atan(tr[1][:,2*i],tr[1][:,2*i-1]) for i in 1:N ]; + + R = reduce(hcat,R); + ϕ = reduce(hcat,ϕ); + return collect(tr[2]),R,ϕ,u0 +end +# ------------------------------------------------ + +# ------------------------------------------------ +""" + simulationLE(;kwargs...) + +Performs a simulations of the system in state and tangent space and provides the Lyapunov exponents. +All arguments are optional, and contains the same arguments as in `simulation` and two additional ones: + * `nLE` number of exponents to be computed. + * `ns` time window between successive calls of the QR-orthogonalization for computation of the exponents. + +Returns: + * `λ` the computed exponents. + * `ds` the DynamicalSystem object at the end of the simulation. +""" +function simulationLE(;K=20.0,p=0.9,α=1.45,gr=erdos_renyi(20, 0.2),tmax=2e2,trans=2e2,nLE=1,ic="random",dt=0.01,ns=1) + N = length(gr.fadjlist) + Γ = 1.0 + ω = tan(α)*(K*cos(α)-Γ) + deg = degree(gr); + ideg = 1.0./deg; + par = K,p,Γ,ω,cos(α),sin(α),gr.fadjlist,ideg,zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),zeros(N),N; + + if typeof(ic)==String || length(ic)!=2*N + if ic=="random" + r0 = rand(N); # random r's + ϕ0 = rand(N)*2*π; + + else + d = 1-2*Γ/(K*cos(α)) + r0 = ( d<0 ? 0.0 : sqrt(d)) + r0 = @. r0*ones(N)+1e-3*randn(N); + r0 = @. abs(r0) + ϕ0 = 1e-3*randn(N) + if ic=="antiphase" + @. ϕ0[1:2:N]=ϕ0[1:2:N]+π + end + end + x0 = @. r0*cos(ϕ0); + y0 = @. r0*sin(ϕ0); + u0 = [ (i%2==1 ? x0[Int(ceil(i/2))] : y0[Int(i/2)] ) for i in 1:2*N]; + else + u0 = ic + end + + # -------------------------------- + # Sparse Jacobian: at least for a ring of N=128 nodes the performance is not noticiably changed, + # but for N = 256 it starts to improve. + if N>=256 + Adj = adjacency_matrix(gr)+I(N) + One = ones(2,2)*1.0 + J0 = Adj⊗One + J0 = sparse(J0) + else + J0=zeros(2*N,2*N) + end + # -------------------------------- + + ds=ContinuousDynamicalSystem(OAnet!, u0, par; + diffeq=(alg=RK4(),dt=dt,adaptive=false)) + tands = TangentDynamicalSystem(ds; k=nLE,J = Jacobian_OAnet!,J0=J0) + + Tdisp=tmax # computation time AFTER transient + trans=trans # transient + + λ = lyapunovspectrum(tands, Int(1*Tdisp/ns); Δt = ns, Ttr=trans) + return λ,ds +end +# ------------------------------------------------ + +# ------------------------------------------------ +""" + statsR(t,R) + +Computes some statistical quantities from the matrix `R` returned by `simulation`. +""" +function statsR(t,R) + n = size(R,2) + nt = size(R,1) + Rs = [ mean(R[:,i]) for i in 1:n ] # mean over time for each population + std_o_time = [ std(R[:,i]) for i in 1:n ] # deviations over time (oscillations or not) + std_o_space = [ std(R[j,:]) for j in 1:nt ] # deviations over space (homogeneous or not) + return mean(Rs),mean(std_o_time),mean(std_o_space) +end +# ------------------------------------------------ + +# ------------------------------------------------ +# Kaplan-York dimension +""" + dimKY(λ;tol=1e-4) + +Computes the Kaplan-York dimension on the exponents in `λ`. +`tol` specifies the threshold at which a exponent with |λ|= 0.0) + if j>0 && j 0.0) + return (j==0 ? 0 : sum(λ[1:j])) +end +# ------------------------------------------------ + +end #module