periodic_orbits.py

#!/usr/bin/env python
# coding: utf-8

# Continuation of Periodic Orbits in the CR3BP
# =====================================
# 
# In this example, we will show how it is possible to use heyoka.py's [expression system](<./The expression system.ipynb>), to compute the state transition matrix of the [circular restricted three-body problem](<./The restricted three-body problem.ipynb>) via [variational equations](<./The variational equations.ipynb>) and outline its use to find periodic orbits via a simple continuation scheme.
# 
# **NOTE**: There is quite some literature on finding periodic orbits in the CR3BP, a plethora of very clever techniques have been developed in the past. This notebook implements only a basic approach as it only aims to show the use of heyoka in this field.
# 
# We make some standard imports:

# In[1]:


import heyoka as hy
import numpy as np
import time

from scipy.optimize import root_scalar

from matplotlib.pylab import plt


# ... and define some functions that will help later on to visualize our trajectories and make nice plots. (ignore them and come back to this later in case you are curious)

# In[2]:


def potential_function(position,mu):
    """Computes the system potential
        Args:
            position (array-like): The position in Cartesian coordinates
            mu (float): The value of the mu parameter.

        Returns:
            The potential
    """
    x,y,z=position
    r_1=np.sqrt((x-mu)**2+y**2+z**2)
    r_2=np.sqrt((x-mu+1)**2+y**2+z**2)
    Omega=1./2.*(x**2+y**2)+(1-mu)/r_1+mu/r_2
    return Omega

def jacobi_constant(state,mu):
    """Computes the system Jacobi constant
        Args:
            state (array-like): The system state (x,y,z,px,py,pz)
            mu (float): The value of the mu parameter.

        Returns:
            The Jacobi constant for the state
    """
    x,y,z,px,py,pz=state
    vx = px + y
    vy = py - x
    vz = pz
    r_1=np.sqrt((x-mu)**2+y**2+z**2)
    r_2=np.sqrt((x-mu+1)**2+y**2+z**2)
    Omega=1/2*(x**2+y**2)+(1-mu)/r_1+mu/r_2
    print(Omega)
    T=1/2*(vx**2+vy**2+vz**2)
    print(T)
    C=Omega-T
    return C


# ## The Circular Restricted 3 Body Problem dynamics

# Let us start defining the equations of motion for the Circular Restricted 3 Body Problem (CR3BP from now on). 
# 
# The problem is usually formulated in a rotating reference frame in which the two massive bodies are at rest. In the rotating reference frame, the equations of motion for the massless particle's cartesian coordinates $\left(x, y, z\right)$ and conjugated momenta $\left(p_x, p_y, p_z\right)$ read:
# 
# $$
# \begin{aligned}
# \dot{x} & = p_x+y,\\
# \dot{y} & = p_y-x, \\
# \dot{z} & = p_z, \\
# \dot{p}_x & = p_y - \frac{1-\mu}{r_{PS}^3}\left( x - \mu \right)-\frac{\mu}{r_{PJ}^3}\left( x - \mu + 1\right), \\
# \dot{p}_y & = -p_x-\left( \frac{1-\mu}{r_{PS}^3} +  \frac{\mu}{r_{PJ}^3}\right)y, \\
# \dot{p}_z & = -\left( \frac{1-\mu}{r_{PS}^3} +  \frac{\mu}{r_{PJ}^3}\right)z,
# \end{aligned}
# $$
# 
# where $\mu$ is a mass parameter, $r_{PS}^2=\left( x-\mu \right)^2+y^2+z^2$ and $r_{PJ}^2=\left( x -\mu + 1 \right)^2+y^2+z^2$. 
# 
# NOTE: In these equations it is assumed that $M_1 + M_2 = 1$ and the Cavendish constant $G=1$. The biggest mass is then indicated with $1-\mu$, while the smallest with $\mu$. The biggest mass is placed in $x = \mu$ and the smallest in $x = \mu-1$ so that the distance between primaries is also 1. All remaining units are induced by these choices.
# 
# We also refer to the whole state with the symbol $\mathbf x = [x,y,z,p_x, p_y, p_z]$ and the right hand side of the dynamic equations with the symbol $\mathbf f$ so that $\dot{\mathbf x} = \mathbf f(\mathbf x)$. In general we use bold for vectors matrices and normal fonts for their components, hence $\mathbf M$ will, as an example, have components $M_{ij}$.
# 
# With respect to the heyoka.py notebook on [circular restricted three-body problem](<./The restricted three-body problem.ipynb>), we will be here making use of numpy arrays of heyoka expressions as to simplify the notation later on when we need to compute the variational equations.

# In[3]:


# Create the symbolic variables.
symbols_state = ["x", "y", "z", "px", "py", "pz"]
x = np.array(hy.make_vars(*symbols_state))
# This will contain the r.h.s. of the equations
f = []

rps_32 = ((x[0] - hy.par[0])**2 + x[1]**2 + x[2]**2)**(-3/2.)
rpj_32 = ((x[0] - hy.par[0]  + 1.)**2 + x[1]**2 + x[2]**2)**(-3/2.)

# The equations of motion.
f.append(x[3] + x[1])
f.append(x[4] - x[0])
f.append(x[5])
f.append(x[4] - (1. - hy.par[0]) * rps_32 * (x[0] - hy.par[0]) - hy.par[0] * rpj_32 * (x[0] - hy.par[0] + 1.))
f.append(-x[3] -((1. - hy.par[0]) * rps_32 + hy.par[0] * rpj_32) * x[1])
f.append(-((1. - hy.par[0]) * rps_32 + hy.par[0] * rpj_32) * x[2])
f = np.array(f)


# Let us also define a function to compute the position of the Lagrangian points:

# In[4]:


# Introduce a compiled function for the evaluation
# of the dynamics equation for px.
px_dyn_cf = hy.cfunc([f[3]], vars=x)

def compute_L_points(mu):
    """Computes The exact position of the Lagrangian points. To do so it finds the zeros of the
    the dynamics equation for px.
    
        Args:
            mu (float): The value of the mu parameter.

        Returns:
            xL1, xL2, xL3, xL45, yL45: The coordinates of the various Lagrangian Points
    """
    # Position of the lagrangian points approximated
    xL1 = (mu-1) + (mu/3/(1-mu))**(1/3)
    xL2 = (mu-1) - (mu/3/(1-mu))**(1/3)
    xL3 = -(mu-1) - 7/12 * mu / (1-mu)
    yL45 = np.sin(60/180*np.pi)
    xL45 = -0.5 + mu

    # Solve for the static equilibrium from the approximated solution
    def equilibrium(expr, x,y):
        return px_dyn_cf([x, y, 0., -y, x, 0.], pars=[mu])[0]
    xL1 = root_scalar(lambda x: equilibrium(f, x,0.), x0=xL1,x1=xL1-1e-2).root
    xL2 = root_scalar(lambda x: equilibrium(f, x,0.), x0=xL2,x1=xL2-1e-2).root
    xL3 = root_scalar(lambda x: equilibrium(f, x,0.), x0=xL3,x1=xL3-1e-2).root;
    return xL1, xL2, xL3, xL45, yL45


# ## The variational equations

# We now compute the variational equations expressing the state transition matrix defined as $\delta \mathbf x(t) = \mathbf \Phi(t)\delta \mathbf x(0)$. We define its $ij$ component as:
# 
# $$
# \Phi_{ij}(t) = \frac{d x_i(t)}{dx_j(0)}
# $$
# 
# hence the variational equations:
# 
# $$
# \frac{d }{dt} \Phi_{ij}(t) = \frac{d}{dt}\left(\frac{d x_i(t)}{dx_j(0)}\right) = \frac{d}{dx_j(0)}\left(\frac{d x_i(t)}{dt}\right) = \frac{d f_i(x(t))}{dx_j(0)}
# $$
# 
# expanding the total derivative in the last term we get:
# 
# $$
# \frac{d}{dt}\Phi_{ij}(t) = \sum_k \frac{\partial f_i}{\partial x_k}\frac{dx_k(t)}{dx_j(0)}=\sum_k \frac{\partial f_i}{\partial x_k} \Phi_{kj}(t)
# $$
# 
# which can be written in compact matrix form as:
# 
# $$
# \frac{d}{dt}\mathbf \Phi(t) = \left[\frac{\partial f_i}{\partial x_k}\right] \mathbf \Phi(t)
# $$
# 
# Note that the initial conditions are, trivially: $\mathbf \Phi(0) = \mathbf I$
# 
# Let us then introduce these variational equations using heyoka.
# 
# First, we define the various symbols for the components of the state transition matrix

# In[5]:


symbols_phi = []
for i in range(6):
    for j in range(6):
        # Here we define the symbol for the variations
        symbols_phi.append("phi_"+str(i)+str(j))  
phi = np.array(hy.make_vars(*symbols_phi)).reshape((6,6))


# Then we find the various $\left[\frac{\partial f_i}{\partial x_k}\right]$:

# In[6]:


dfdx = []
for i in range(6):
    for j in range(6):
        dfdx.append(hy.diff(f[i],x[j]))
dfdx = np.array(dfdx).reshape((6,6))


# ... and finally the r.h.s. of the variational equations is:

# In[7]:


# The (variational) equations of motion
dphidt = dfdx@phi


# how very very beautiful!
# 
# **NOTE**: The variational equations are here written for a chaotic system, thus for long interation times the variations can explode and have a negative influence for the step size control. Nothing we can do about it, it's chaos! 

# ## Putting all together and integrating some initial conditions
# Let us put all the equations 6 + 6x6 = 42 into one big Taylor integrator and perform one numerical integration.
# 
# First, we create the dynamics ...

# In[8]:


dyn = []
for state, rhs in zip(x,f):
    dyn.append((state, rhs))
for state, rhs in zip(phi.reshape((36,)),dphidt.reshape((36,))):
    dyn.append((state, rhs))
# These are the initial conditions on the variational equations (the identity matrix)
ic_var = np.eye(6).reshape((36,)).tolist()


# ... then we instantiate the Taylor integrator (high accuracy and no compact mode)

# In[9]:


start_time = time.time()
ta = hy.taylor_adaptive(
    # The ODEs.
    dyn,
    # The initial conditions.
    [-0.45, 0.80, 0.00, -0.80, -0.45, 0.58] + ic_var,
    # Operate below machine precision
    # and in high-accuracy mode.
    tol = 1e-18, high_accuracy = True
)
print("--- %s seconds --- to build the Taylor integrator" % (time.time() - start_time))


# ... and we perform and time a numerical propagation for these conditions

# In[10]:


ic = [-0.72, 0.0, 0, 0.0, -0.2, 0.]
t_final=200
mu=0.01
ta.pars[0] = mu
# Reset the state
ta.time = 0
ta.state[:] = ic + ic_var
# Time grid
t_grid = np.linspace(0, t_final, 2000)


# In[11]:


# Go ...
start_time = time.time()
out = ta.propagate_grid(t_grid)
print("--- %s seconds --- to propagate" % (time.time() - start_time))


# To check and validate, let's plot the trajectory and some cosmetics to visualize the solution

# In[12]:


plt.figure(figsize=(14,14))

# Lets find the postiion of the lagrangian points
xL1, xL2, xL3, xL45, yL45 = compute_L_points(mu)
# We also compute the Jacobi constant
C_jacobi = jacobi_constant(ic, mu)

# Plot the trajectory (xy)
plt.subplot(1,2,1)
plt.plot(out[5][:, 0], out[5][:, 1])
plt.xlabel("x")
plt.ylabel("y")
# Plot the zero velocity curve
xx = np.linspace(-1.5,1.5,2000)
yy = np.linspace(-1.5,1.5,2000)
x_grid,y_grid = np.meshgrid(xx,yy)
im = plt.imshow( ((potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu)<=C_jacobi)).astype(int) , 
                extent=(x_grid.min(),x_grid.max(),y_grid.min(),y_grid.max()),origin="lower", cmap="Greens")
# Plot the lagrangian points and primaries
plt.scatter(mu, 0, c='k', s=300)
plt.scatter(mu-1, 0, c='k', s=150)
plt.scatter(xL1, 0, c='r')
plt.scatter(xL2, 0, c='r')
plt.scatter(xL3, 0, c='r')
plt.scatter(-0.5+mu, yL45, c='r')
plt.scatter(-0.5+mu, -yL45, c='r')

# Plot the trajectory (xz)
plt.subplot(1,2,2)
plt.plot(out[5][:, 0], out[5][:, 2])
plt.xlabel("x")
plt.ylabel("z");
# Plot the zero velocity curve
xx = np.linspace(-1.5,1.5,2000)
zz = np.linspace(-1.5,1.5,2000)
x_grid,z_grid = np.meshgrid(xx,zz)
im = plt.imshow( ((potential_function((x_grid,np.zeros(np.shape(x_grid)), z_grid),mu=mu)<=C_jacobi)).astype(int) , 
                extent=(x_grid.min(),x_grid.max(),z_grid.min(),z_grid.max()),origin="lower", cmap="Greens")
# Plot the lagrangian points and primaries
plt.scatter(mu, 0, c='k', s=300)
plt.scatter(mu-1, 0, c='k', s=150)
plt.scatter(xL1, 0, c='r')
plt.scatter(xL2, 0, c='r')
plt.scatter(xL3, 0, c='r')
plt.scatter(-0.5+mu, 0, c='r')
plt.scatter(-0.5+mu, 0, c='r')


# In[13]:


np.min(potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu))


# In[14]:


C_jacobi


# All fine ..... at least visually! So far we have not made use of the variational equations at all, but this is about to change!

# ## Finding Periodic Orbits
# To find a periodic orbit in a dynamical system, a first step to then possibly find a whole family of them, we will proceed as follows:
# 
# * Get some *decent* initial conditions, for example one can use the [Poincaré–Lindstedt method](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Lindstedt_method) or, in the specific case of the CR3BP, the work from Richardson).
# 
# Richardson, D. L. (1980). Analytic construction of periodic orbits about the collinear points. Celestial mechanics, 22(3), 241-253.
# 
# * Once some initial guess $\mathbf x_0$ is available for the initial state and $T$ for the period, we write the Taylor first order expansion of the system solution as:
# 
# $$
# \mathbf x = \overline {\mathbf x} + \mathbf \Phi \delta \mathbf x_0 + \mathbf \Phi_T \delta T
# $$
# 
# where $\mathbf \Phi = \left[\frac{\partial x_i}{\partial x_{0_k}}\right] $ is computed via the variational equations, $\mathbf \Phi_T = \left[\frac{\partial x_i}{\partial t}\right] = \dot{\mathbf x} = \mathbf f$ and $\overline {\mathbf x}$ is the final state reached starting from $\mathbf x_0$ and integrating for $T$. Such an expansion tells us how much the state evaluated in $T$ would change if we move the initial conditions by $\delta\mathbf x_0$ and the integration time by $\delta T$. 
# 
# * Now (**pay attention, as here is the whole trick**), we write the periodicity condition enforcing that after $T+\delta T$ the state goes back to $\mathbf x_0 + \delta \mathbf x_0$:
# 
# $$
# \overline {\mathbf x} + \mathbf \Phi \delta \mathbf x_0 + \mathbf f \delta T = \mathbf x_0 + \delta \mathbf x_0
# $$
# 
# which is rearranged in the form:
# 
# $$
# \left(\mathbf \Phi -\mathbf I\right) \delta \mathbf x_0 + \mathbf f \delta T = \mathbf x_0 -\overline {\mathbf x}
# $$
# 
# This fundamental relation is at the basis of any numerical algorithm that wants to find a closed periodic orbit. It is a system of 6 equation in the 7 unknowns $\delta \mathbf x, \delta T$: as a consequence, it is overdetermined. We then must choose among the infinitely many solutions one. We do so adding the Poincare' phasing condition, which requests that:
# 
# $$
# \mathbf f \cdot \delta \mathbf x_0 =  \mathbf 0
# $$
# in other word we will restrict our $\delta x$ to the hyperplane plane perpendicular to the dynamics.
# 
# We now have seven equations and seven unknowns. Seems like we are done (as far as $\mathbf \Phi -\mathbf I$ has full rank). 
# 
# Let us implement a naive iterative scheme and close some orbit. 
# 
# First we play to find a decent initial condition ....

# In[15]:


# New mu parameter (no reason to change, just came out playing)
mu = 0.01215057
# Initial guess for the integration time (will eventually converge to a period)
t_final = 3.
# We recomupte the lagrangian points
xL1, xL2, xL3, xL45, yL45 = compute_L_points(mu)

# Initial conditions in the cartesian representation x,y,z,vx,vy,vz
ic_cart = [ -8.36809444e-01, 0.,0.,0.,  -8.85435468e-04, 0.]
# Now in conj_mom
ic = [ic_cart[0], ic_cart[1], ic_cart[2], ic_cart[3]- ic_cart[1], ic_cart[4] + ic_cart[0], ic_cart[5]]
# We recompute  the Jacobi constant
C_jacobi = jacobi_constant(ic, mu)

# Reset the state
ta.time = 0.
ta.state[:] = ic + ic_var
ta.pars[0] = mu
# Time grid
t_grid = np.linspace(0, t_final, 2000)
# Go ...
out0 = ta.propagate_grid(t_grid)


# In[16]:


np.abs(xL1 -  -8.36809444e-01) 


# In[17]:


C_jacobi


# In[18]:


potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu)


# We plot the initial orbit (zooming in the Lagrangian point)

# In[19]:


plt.figure(figsize=(9,9))

plt.subplot(1,1,1)
zoom=0.005
plt.plot(out0[5][:, 0], out0[5][:, 1],'r')

plt.xlabel("x")
plt.ylabel("y")

# Plot the zero velocity curve
xx = np.linspace(xL1-zoom,xL1+zoom,2000)
yy = np.linspace(-zoom,zoom,2000)
x_grid,y_grid = np.meshgrid(xx,yy)
im = plt.imshow( ((potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu)<=C_jacobi)).astype(int) , 
                extent=(x_grid.min(),x_grid.max(),y_grid.min(),y_grid.max()),origin="lower", cmap="Greens")

# Plot the lagrangian points and primaries
plt.scatter(mu, 0, c='k', s=300)
plt.scatter(mu-1, 0, c='k', s=100)
plt.scatter(xL1, 0, c='r')
plt.scatter(xL2, 0, c='r')
plt.scatter(xL3, 0, c='r')
plt.scatter(-0.5+mu, yL45, c='r')
plt.scatter(-0.5+mu, -yL45, c='r')


plt.xlim(xL1-zoom, xL1+zoom)
plt.ylim(-zoom, +zoom);


# The orbit is good but does not close! Let us build an iteration that corrects $x_0,y_0, z_0, p_{x_0}, p_{y_0}, p_{z_0}, T$ as to close the orbit.

# In[20]:


# Introduce a compiled function for the evaluation
# of the dynamics equations.
dyn_cf = hy.cfunc(f, vars=x)

def corrector(ta, x0):
    """
    Performs and logs a step of a corrector algorithm that takes a numerical integration from x0 -> T -> xf. The result
    is a new tentative x0 that should result in a closed orbit.
    """
    x0 = np.array(x0)
    mu = ta.pars[0]
    t_final = ta.time
    
    state_T = ta.state[:6]
    
    Phi = ta.state[6:].reshape((6,6))
    dynT = dyn_cf(state_T, pars=[mu]).reshape((-1,1))

    # We add as last state delta T
    A = np.concatenate((Phi-np.eye(6),dynT), axis=1)
    # We add the Poincare phasing condition as a last equation
    phasing_cond = np.insert(dynT,-1,0).reshape((1,-1))

    A = np.concatenate((A, phasing_cond))
    # We construct the r.h.s.
    b = (x0 - state_T).reshape(-1,1)
    print("error was:", np.linalg.norm(b))
    # need to add the zero corresponding to the phasing condition
    b = np.insert(b,-1,0)
    
    delta = np.linalg.inv(A)@b
    print("condition number is:", np.linalg.cond(A))
    
    x0_new = x0+delta[:6]
    t_final = t_final+delta[-1]

    # Reset the state
    ta.time = 0.
    ta.state[:] = x0_new.reshape((-1,)).tolist() + ic_var
    # Go ...
    ta.propagate_until(t_final)
    # New error is:
    b = (x0_new - ta.state[:6]).reshape(-1,1)
    print("new error is:", np.linalg.norm(b))
    return ta, x0_new.tolist()


# Lets do some corrector iterations (not too many since as we get near to a periodic orbit, the condition number will explode)

# In[21]:


ic_periodic = ic
for i in range(6):
    ta, ic_periodic = corrector(ta, ic_periodic)


#  .... et voila'!! As expected the iterations converge to a periodic orbit, while the matrix condition number increases to infinite as $\mathbf \Phi$ becomes a monodromy matrix.
#  
# Of course, we now visualize the orbit as to make sure its closed!

# In[22]:


t_final = ta.time

# We compute  the IC Jacobi constant
C_jacobi = jacobi_constant(ic_periodic, mu)

# Reset the state
ta.time = 0.
ta.state[:] = ic_periodic + ic_var
ta.pars[0] = mu
# Time grid
t_grid = np.linspace(0, t_final, 2000)
# Go ...
out = ta.propagate_grid(t_grid)


# In[23]:


# We plot the initial condition (zoom in the Lagrangian point)
plt.figure(figsize=(9,9))

plt.subplot(1,1,1)
zoom=0.005
plt.plot(out0[5][:, 0], out0[5][:, 1],'r')
plt.plot(out[5][:, 0], out[5][:, 1],'k')

plt.xlabel("x")
plt.ylabel("y")

# Plot the zero velocity curve
xx = np.linspace(xL1-zoom,xL1+zoom,2000)
yy = np.linspace(-zoom,zoom,2000)
x_grid,y_grid = np.meshgrid(xx,yy)
im = plt.imshow( ((potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu)<=C_jacobi)).astype(int) , 
                extent=(x_grid.min(),x_grid.max(),y_grid.min(),y_grid.max()),origin="lower", cmap="Greens")

# Plot the lagrangian points and primaries
plt.scatter(mu, 0, c='k', s=300)
plt.scatter(mu-1, 0, c='k', s=100)
plt.scatter(xL1, 0, c='r')
plt.scatter(xL2, 0, c='r')
plt.scatter(xL3, 0, c='r')
plt.scatter(-0.5+mu, yL45, c='r')
plt.scatter(-0.5+mu, -yL45, c='r')


plt.xlim(xL1-zoom, xL1+zoom)
plt.ylim(-zoom, +zoom)

print("mu: ", ta.pars[0])
print(f"Initial condition: [{out[5][0,0]:.16e}, {out[5][0,1]:.16e}, {out[5][0,2]:.16e}, {out[5][0,3]:.16e}, {out[5][0,4]:.16e}, {out[5][0,5]:.16e}]")
print(f"Period: {t_final:.16e}")


# In[24]:


out


# ## Continuing into a family of periodic orbits.
# 
# Since we now have initial conditions $\mathbf x_0$ that result in a periodic orbit, necessarily $\mathbf x_0 =\overline {\mathbf x}$, hence the periodicity condition becomes:
# 
# $$
# \left(\mathbf \Phi -\mathbf I\right) \delta \mathbf x_0 + \mathbf f \delta T = \mathbf 0
# $$
# 
# which is a system of 6 equations in seven unknowns. Futhermore, the monodromy matrix $\mathbf \Phi$ has now the eigenvalue 1, and thus $\left(\mathbf \Phi -\mathbf I\right)$ is not invertible! 
# 
# This corresponds, physically, to the fact that there are infinite possibilities to choose  $\delta \mathbf x_0$ so that $\mathbf x_0 + \delta \mathbf x_0$ results in a new periodic orbit having period $T + \delta T$.
# 
# We still make use of the Poincare' phasing condition so that the overall system we actually consider is:
# 
# $$
# \left\{
# \begin{array}{c}
# \left(\mathbf \Phi -\mathbf I\right) \delta \mathbf x_0 + \mathbf f \delta T = \mathbf 0\\
# \mathbf f \cdot \delta \mathbf x_0 =  0
# \end{array}
# \right.
# $$
# 
# which we write as:
# 
# $$
# \mathbf A_f \delta {\mathbf x_f} = \mathbf 0
# $$
# 
# where we considered the full state $\delta {\mathbf x_f}$ including both $\delta \mathbf x_0$ and $\delta T$. The rank of the square 7x7 full matrix ${\mathbf A_f}$ is only 6, thus the linear system admits non trivial solutions. To find them we fix the value of one component of $\delta {\mathbf x_f}$ and thus obtain a new system with a reduced state and matrix:
# 
# $$
# \mathbf A_r\delta {\mathbf x_r} = \mathbf b
# $$
# 
# This is an overdetermined system of seven equations in six variables, but having only one only solution which we find as:
# 
# $$
# \delta \mathbf x_r = (\mathbf A^T \mathbf A)^{-1}\mathbf A^T\mathbf b
# $$
# 
# We have thus found a new initial guess to start with to find the next closed orbit in the family.

# To start with, as validation and test of what we got so far, we get the monodromy matrix (from the last numerical integrated periodic orbit) ... 

# In[25]:


Phi = ta.state[6:].reshape((6,6))


# And compute its eigenvalues

# In[26]:


eigv = np.linalg.eigvals(Phi)
print(eigv)


# As expecetd we have two eigenvalues $\lambda_{3,4}$ equal to one, two real ones so that $\lambda_{1}\cdot \lambda_2=1$ (stable and unstable manifold) and two complex conjugated ones $\lambda_{5}\cdot \lambda_6=1$.
# 
# We write a simple function that takes as argument the result of some numerical integration over periodic conditions and creates new tentative initial guess continuing on the state parameter indexed by *idx*, (defaults to $x$).
# 
# Note that in the following function we assemble the full matrix $\mathbf A_f$ completed with the Poincare phase condition and having we consider as state ${\mathbf x_f} = [\delta x,\delta y,\delta z,\delta p_x,\delta p_y,\delta p_z,\delta T]$.  We then select a column, fix the corresponding value for the state variable, bring it to the right side and solve the resulting system which is overdetermined, but admits a unique solution since $\mathbf \Phi - \mathbf I$ is singular.

# In[27]:


def predictor(ta, idx=0, variation=1e-4):
    Phi = ta.state[6:].reshape((6,6))
    state_T = ta.state[:6]
    state_T_dict = {"x":state_T[0], "y":state_T[1], "z":state_T[2], "px":state_T[3], "py":state_T[4], "pz":state_T[5]}
    # Compute the dynamics from its expressions
    dynT = dyn_cf(state_T, pars=[mu]).reshape((-1,1))
    # Computing the full A
    A = np.concatenate((Phi-np.eye(6), dynT.T))
    fullA = np.concatenate((A,np.insert(dynT,-1,0).reshape((-1,1))), axis=1)
    # Computing the A resulting from fixing the continuation parameter to a selected state.
    A = fullA[:,list(set(range(7))-set([idx]))]
    b = - fullA[:,[idx]] * variation
    # We solve.
    dx = np.linalg.inv((A.T@A)) @ (A.T@b)
    # Assembling back the full state (x,y,z,px,py,pz,T)
    dx = np.insert(dx,idx,variation)
    return dx


# We can now use the function to create a new initial guess:

# In[28]:


dx = predictor(ta)
ic_continued_guess = [a+b for a,b in zip(ic_periodic, dx[:6].tolist())]
new_T = ta.time + dx[-1]


# Let us visualize as usual the resulting orbit .... (code is repeated for convenience, but its identical to above cells)

# In[29]:


C_jacobi = jacobi_constant(ic_continued_guess, mu)

# Reset the state
ta.time = 0.
ta.state[:] = ic_continued_guess + ic_var
ta.pars[0] = mu
# Time grid
t_grid = np.linspace(0, new_T, 2000)
# Go ...
out2 = ta.propagate_grid(t_grid)


# In[30]:


# We plot the initial condition (zoom in the Lagrangian point)
plt.figure(figsize=(9,9))

plt.subplot(1,1,1)
zoom=0.005
plt.plot(out0[5][:, 0], out0[5][:, 1], 'r')
plt.plot(out[5][:, 0], out[5][:, 1],'k')
plt.plot(out2[5][:, 0], out2[5][:, 1],'r')

plt.xlabel("x")
plt.ylabel("y")

# Plot the zero velocity curve
xx = np.linspace(xL1-zoom,xL1+zoom,2000)
yy = np.linspace(-zoom,zoom,2000)
x_grid,y_grid = np.meshgrid(xx,yy)
im = plt.imshow( ((potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu)<=C_jacobi)).astype(int) , 
                extent=(x_grid.min(),x_grid.max(),y_grid.min(),y_grid.max()),origin="lower", cmap="Greens")

# Plot the lagrangian points and primaries
plt.scatter(mu, 0, c='k', s=300)
plt.scatter(mu-1, 0, c='k', s=100)
plt.scatter(xL1, 0, c='r')
plt.scatter(xL2, 0, c='r')
plt.scatter(xL3, 0, c='r')
plt.scatter(-0.5+mu, yL45, c='r')
plt.scatter(-0.5+mu, -yL45, c='r')


plt.xlim(xL1-zoom, xL1+zoom)
plt.ylim(-zoom, +zoom)


# It's nearly closed, but not that well .... it will need a correction ... but we can call the same iterations we made previously when we closed the first initial guess, remember?

# In[31]:


ic_continued = ic_continued_guess
for i in range(3):
    ta, ic_continued = corrector(ta, ic_continued)


# And we visualize all the initial guesses and closed orbit found so far:

# In[32]:


t_final = ta.time

# We compute  the IC Jacobi constant
C_jacobi = jacobi_constant(ic_continued, mu)

# Reset the state
ta.time = 0.
ta.state[:] = ic_continued + ic_var
ta.pars[0] = mu
# Time grid
t_grid = np.linspace(0, t_final, 2000)
# Go ...
out3 = ta.propagate_grid(t_grid)


# In[33]:


# We plot the initial condition (zoom in the Lagrangian point)
plt.figure(figsize=(9,9))

plt.subplot(1,1,1)
zoom=0.005
plt.plot(out0[5][:, 0], out0[5][:, 1],'r')
plt.plot(out[5][:, 0], out[5][:, 1],'k')
plt.plot(out2[5][:, 0], out2[5][:, 1],'r')
plt.plot(out3[5][:, 0], out3[5][:, 1],'k')

plt.xlabel("x")
plt.ylabel("y")

# Plot the zero velocity curve
xx = np.linspace(xL1-zoom,xL1+zoom,2000)
yy = np.linspace(-zoom,zoom,2000)
x_grid,y_grid = np.meshgrid(xx,yy)
im = plt.imshow( ((potential_function((x_grid,y_grid,np.zeros(np.shape(x_grid))),mu=mu)<=C_jacobi)).astype(int) , 
                extent=(x_grid.min(),x_grid.max(),y_grid.min(),y_grid.max()),origin="lower", cmap="Greens")

# Plot the lagrangian points and primaries
plt.scatter(mu, 0, c='k', s=300)
plt.scatter(mu-1, 0, c='k', s=100)
plt.scatter(xL1, 0, c='r')
plt.scatter(xL2, 0, c='r')
plt.scatter(xL3, 0, c='r')
plt.scatter(-0.5+mu, yL45, c='r')
plt.scatter(-0.5+mu, -yL45, c='r')


plt.xlim(xL1-zoom, xL1+zoom)
plt.ylim(-zoom, +zoom);


# Found it! In order to do this in a loop, though, we will need to do better as eventually this will fail as the continuation parameter
# here used, i.e., the period $T$, will create issues as it is not monotonically increasng with the orbit size.
# 
# The solution requires a bit more math, and it is given by the definition and use of a [pseudo arc-length](<./Pseudo arc-length continuation in the CR3BP.ipynb>).