The Lorenz attractor


The Lorenz system of coupled, ordinary, first-order differential equations have chaotic solutions for certain parameter values $\sigma$, $\rho$ and $\beta$ and initial conditions, $u(0)$, $v(0)$ and $w(0)$.

\begin{align*} \frac{\mathrm{d}u}{\mathrm{d}t} &= \sigma (v - w)\\ \frac{\mathrm{d}v}{\mathrm{d}t} &= \rho u - v - uw\\ \frac{\mathrm{d}w}{\mathrm{d}t} &= uv - \beta w \end{align*}

The following program plots the Lorenz attractor (the values of $x$, $y$ and $z$ as a parametric function of time) on a Matplotlib 3D projection.

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Lorenz paramters and initial conditions
sigma, beta, rho = 10, 2.667, 28
u0, v0, w0 = 0, 1, 1.05

# Maximum time point and total number of time points
tmax, n = 100, 10000

def lorenz(X, t, sigma, beta, rho):
    """The Lorenz equations."""
    u, v, w = X
    up = -sigma*(u - v)
    vp = rho*u - v - u*w
    wp = -beta*w + u*v
    return up, vp, wp

# Integrate the Lorenz equations on the time grid t
t = np.linspace(0, tmax, n)
f = odeint(lorenz, (u0, v0, w0), t, args=(sigma, beta, rho))
x, y, z = f.T

# Plot the Lorenz attractor using a Matplotlib 3D projection
fig = plt.figure()
ax = fig.gca(projection='3d')

# Make the line multi-coloured by plotting it in segments of length s which
# change in colour across the whole time series.
s = 10
c = np.linspace(0,1,n)
for i in range(0,n-s,s):
    ax.plot(x[i:i+s+1], y[i:i+s+1], z[i:i+s+1], color=(1,c[i],0), alpha=0.4)

# Remove all the axis clutter, leaving just the curve.

The Lorenz attractor

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