TP.py

import numpy as np
from numpy.linalg import qr, solve, norm
from scipy.linalg import expm
from rossler_map import RosslerMap

import pandas as pd
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt

def lyapunov_exponent(traj, jacobian, max_it=1000, delta_t=1e-3):

    n = traj.shape[1]
    w = np.eye(n)
    rs = []
    chk = 0

    for i in range(max_it):
        jacob = jacobian(traj[i,:])
        #WARNING this is true for the jacobian of the continuous system!
        w_next = np.dot(expm(jacob * delta_t),w)
        #if delta_t is small you can use:
        #w_next = np.dot(np.eye(n)+jacob * delta_t,w)

        w_next, r_next = qr(w_next)

        # qr computation from numpy allows negative values in the diagonal
        # Next three lines to have only positive values in the diagonal
        d = np.diag(np.sign(r_next.diagonal()))
        w_next = np.dot(w_next, d)
        r_next = np.dot(d, r_next.diagonal())

        rs.append(r_next)
        w = w_next
        if i//(max_it/100)>chk:
            print(i//(max_it/100))
            chk +=1

    return  np.mean(np.log(rs), axis=0) / delta_t

def newton(f,jacob,x):
    #newton raphson method
    tol =1
    while tol>1e-5:
        #WARNING this is true for the jacobian of the continuous system!
        tol = x
        x = x-solve(jacob(x),f(v=x))
        tol = norm(tol-x)
    return x



if __name__ == '__main__':

    Niter = 2000000

    delta_t = 1e-2
    ROSSLER_MAP = RosslerMap(delta_t=delta_t)
    INIT = np.array([-5.75, -1.6,  0.02])
    traj,t = ROSSLER_MAP.full_traj(Niter, INIT)

    fig = plt.figure()
    ax = fig.gca(projection='3d')
    ax.plot(traj[:,0], traj[:,1], traj[:,2])

    fix_point = newton(ROSSLER_MAP.v_eq,ROSSLER_MAP.jacobian,INIT)

    error = norm(fix_point-ROSSLER_MAP.equilibrium())
    print("equilibrium state :", fix_point, ", error : ", error)

    #lyap = lyapunov_exponent(traj, ROSSLER_MAP.jacobian, max_it=Niter, delta_t=delta_t)
    #print("Lyapunov Exponents :", lyap, "with delta t =", delta_t)

    plt.show()

    df = pd.DataFrame({"x":traj[:,0],"y":traj[:,1], "z":traj[:,2]})
    df.to_csv("data.csv")