Lorenz Attractor
The resulting equations for the time dependent parameters of the approximate Lorenz convection equations are
Where
with the critical Rayleigh number
Question
1 - Identify
as
and write a python (or another programming language ) to solve the time evolution of the three variables.
2 - Change
to 10, and then 24. Plot the time evolution of the variables and comment on how these solutions differ.
Solution
1
"""2
@author: yacine.mezemate3
"""4
from __future__ import division
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import numpy as np
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from scipy.integrate import odeint
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import matplotlib.pyplot as plt
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from mpl_toolkits.mplot3d import Axes3D
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from matplotlib.colors import cnames
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# System of differential equations16
def LorenzSystem((x,y,z), t0, sigma=10., beta=8./3, rho=28.0):
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return [sigma*(y-x), x*(rho-z)-y, x*y -beta*z]
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#Initial condition20
np.random.seed(1)
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x0 = -15 + 30 * np.random.random((30, 3))
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# Time 24
t = np.linspace(0, 10, 1000)
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#System resolution27
x_t = np.asarray([odeint(LorenzSystem,x0i,t) for x0i in x0])
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# Plots31
fig = plt.figure()
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ax = fig.add_subplot(111, projection='3d')
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for i in range (0,29):
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ax.plot_wireframe(x_t[i,:,0],x_t[i,:,1],x_t[i,:,2])
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plt.title("Lorenz attractor", fontsize =18)
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ax.set_xlabel('X', fontsize =18)
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ax.set_ylabel('Y', fontsize =18)
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ax.set_zlabel('Z', fontsize =18)
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plt.figure(2)
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plt.plot(t,x_t[0,:,0], label="X(t)")
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plt.plot(t,x_t[0,:,1], label="Y(t)")
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plt.plot(t,x_t[0,:,2], label="Z(t)")
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plt.xlabel("time", fontsize=18)
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plt.ylabel("variables", fontsize =18)
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plt.legend()
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plt.show()
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