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# Load a custom CSS and import plotting
# https://github.com/fperez/pycon2014-keynote
%run static/talktools/talktools.py
#makes sure inline plotting is enabled
%pylab inline 
#set figure size
figsize(20, 6)









    












    



Populating the interactive namespace from numpy and matplotlib

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.



In [2]:

    
%matplotlib inline
from IPython.html.widgets import interact, interactive
from IPython.display import clear_output, display, HTML
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

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 Lorentz system."""
        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



In [3]:

    
t, x_t = solve_lorenz(angle=0, N=10)



In [4]:

    
w = interactive(
            solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0)
            )
display(w)



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