Lorenz Attractor

by Paulo Marques, 2013/09/21


This notebook implements the beautiful Lorenz Attractor in Python. The Lorenz Attractor is probably the most ilustrative example of a system that exibits cahotic behaviour. Slightly changing the initial conditions of the system leads to completely different solutions. The system itself corresponds to the movement of a point particle in a 3D space over time.

![Lorenz Attractor](http://upload.wikimedia.org/wikipedia/commons/e/e0/Lorenz.png)

The system is formally described by three different differential equations. These equations represent the movement of a point $(x, y, z)$ in space over time. In the following equations, $t$ represents time, $\sigma$, $\rho$, $\beta$ are constants.

$$ \frac{dx}{dt} = \sigma (y - x) $$

$$ \frac{dy}{dt} = x (\rho - z) - y $$

$$ \frac{dz}{dt} = x y - \beta z $$

Let's implement it in python.


Let's start by importing some basic libraries.

In [1]:
%pylab inline
from scipy.integrate import odeint
from mpl_toolkits.mplot3d.axes3d import Axes3D
Populating the interactive namespace from numpy and matplotlib

We need to define the system of differential equations as an equation of the form: ${\bf r}' = {\bf f}({\bf r},t)$ where ${\bf r} = (x, y, z)$ and ${\bf f}({\bf r},t)$ is the mapping function.

In [2]:
def f(r, t):
    (x, y, z) = r
    
    # The Lorenz equations
    dx_dt = sigma*(y - x)
    dy_dt = x*(rho - z) - y
    dz_dt = x*y - beta*z
    
    return [dx_dt, dy_dt, dz_dt]    

Let's define the initial conditions of the system ${\bf r}_0 = (x_0, y_0, z_0)$, the constants $\sigma$, $\rho$ and $\beta$ and a time grid.

In [3]:
# Initial position in space
r0 = [0.1, 0.0, 0.0]

# Constants sigma, rho and beta
sigma = 10.0
rho   = 28.0
beta  = 8.0/3.0

# Time grid
tf = 100.0
t = linspace(0, tf, tf*100)

Now let's solve the differencial equations numericaly and extract the corresponding $(x, y, z)$:

In [4]:
pos = odeint(f, r0, t)

x = pos[:, 0]
y = pos[:, 1]
z = pos[:, 2]

Let's see how it looks in 3D.

In [5]:
fig = figure(figsize=(16,10))
ax = fig.gca(projection='3d')
ax.plot(x, y, z)
Out[5]:
[<mpl_toolkits.mplot3d.art3d.Line3D at 0x111a61bd0>]

Let's see different cuts around the axes:

In [6]:
fig, ax = subplots(1, 3, sharex=True, sharey=True, figsize=(16,8))

ax[0].plot(x, y)
ax[0].set_title('X-Y cut')

ax[1].plot(x, z)
ax[1].set_title('X-Z cut')

ax[2].plot(y, z)
ax[2].set_title('Y-Z cut')
Out[6]:
<matplotlib.text.Text at 0x112197a10>

MIT LICENSE

Copyright (C) 2013 Paulo Marques (pjp.marques@gmail.com)

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

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