#!/usr/bin/env python # coding: utf-8 # # Exploring the Lorenz System of Differential Equations # In this Notebook we explore the Lorenz system of differential equations: # # # \begin{aligned} # \dot{x} & = \sigma(y-x) \\ # \dot{y} & = \rho x - y - xz \\ # \dot{z} & = -\beta z + xy # \end{aligned} # # # This is one of the classic systems in non-linear differential equations. It exhibits a range of different behaviors as the parameters ($\sigma$, $\beta$, $\rho$) are varied. # ## Imports # First, we import the needed things from IPython, NumPy, Matplotlib and SciPy. # In[ ]: get_ipython().run_line_magic('matplotlib', 'inline') # In[ ]: from IPython.html.widgets import interact, interactive from IPython.display import clear_output, display, HTML # In[ ]: import numpy as np from scipy import integrate from matplotlib import pyplot as plt from mpl_toolkits.mplot3d import Axes3D from matplotlib.colors import cnames from matplotlib import animation # ## Computing the trajectories and plotting the result # We define a function that can integrate the differential equations numerically and then plot the solutions. This function has arguments that control the parameters of the differential equation ($\sigma$, $\beta$, $\rho$), the numerical integration (N, max_time) and the visualization (angle). # In[ ]: def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0): fig = plt.figure() ax = fig.add_axes([0, 0, 1, 1], projection='3d') ax.axis('off') # prepare the axes limits ax.set_xlim((-25, 25)) ax.set_ylim((-35, 35)) ax.set_zlim((5, 55)) def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho): """Compute the time-derivative of a Lorenz system.""" x, y, z = x_y_z return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z] # Choose random starting points, uniformly distributed from -15 to 15 np.random.seed(1) x0 = -15 + 30 * np.random.random((N, 3)) # Solve for the trajectories t = np.linspace(0, max_time, int(250*max_time)) x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t) for x0i in x0]) # choose a different color for each trajectory colors = plt.cm.jet(np.linspace(0, 1, N)) for i in range(N): x, y, z = x_t[i,:,:].T lines = ax.plot(x, y, z, '-', c=colors[i]) plt.setp(lines, linewidth=2) ax.view_init(30, angle) plt.show() return t, x_t # Let's call the function once to view the solutions. For this set of parameters, we see the trajectories swirling around two points, called attractors. # In[ ]: t, x_t = solve_lorenz(angle=0, N=10) # Using IPython's interactive function, we can explore how the trajectories behave as we change the various parameters. # In[ ]: w = interactive(solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0)) display(w) # The object returned by interactive is a Widget object and it has attributes that contain the current result and arguments: # In[ ]: t, x_t = w.result # In[ ]: w.kwargs # After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in $x$, $y$ and $z$. # In[ ]: xyz_avg = x_t.mean(axis=1) # In[ ]: xyz_avg.shape # Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors. # In[ ]: plt.hist(xyz_avg[:,0]) plt.title('Average $x(t)$') # In[ ]: plt.hist(xyz_avg[:,1]) plt.title('Average $y(t)$')