No book on scientific computing is complete without revisiting Lorenz attractors; SciPy excels both at computation of solutions and presentation of ideas based upon systems of differential equations, of course, and we will show how and why in this section.

Consider a two-dimensional fluid cell that is heated from underneath and cooled from above, much like what occurs with the Earth's atmosphere. This creates convection that can be modeled by a single partial differential equation, for which a decent approximation has the form of the following system of ordinary differential equations:

The variable x represents the rate of convective overturning. The variables y and z stand for the horizontal and vertical temperature variations, respectively. This system depends on four physical parameters, the descriptions of which are far beyond the scope of this book. The important point is that we may model Earth's atmosphere with these equations, and in that case a good choice for the parameters is given by sigma = 10.0, and b = 8/3.0. For certain values of the third parameter, we have systems for which the solutions behave chaotically. Let's explore this effect with the help of SciPy.

In the following code snippet, we will use one of the solvers in the scipy.integrate module as well as the plotting utilities:

>>> import numpy
>>> from numpy import linspace
>>> import scipy
>>> from scipy.integrate import odeint
>>> import matplotlib.pyplot as plt
>>> from mpl_toolkits.mplot3d import Axes3D
>>> sigma=10.0
>>> b=8/3.0
>>> r=28.0
>>> f = lambda x,t: [sigma*(x[1]-x[0]), r*x[0]-x[1]-x[0]*x[2], x[0]*x[1]-b*x[2]]

Let's choose a time interval t large enough with a sufficiently dense partition and any initial condition, y0. Then, issue the following commands:

>>> t=linspace(0,20,2000); y0=[5.0,5.0,5.0]
>>> solution=odeint(f,y0,t)
>>> X=solution[:,0]; Y=solution[:,1]; Z=solution[:,2]

If we want to plot a 3D rendering of the solution obtained, we can do so as follows:

>>> import matplotlib.pyplot as plt
>>> plt.gca(projection='3d'), plt.plot(X,Y,Z)
>>> plt.show()

This produces the following graph that shows a Lorenz attractor:

This is most illustrative and shows precisely the chaotic behavior of the solutions. Let's observe the fluctuations of the vertical temperature in detail, along with the fluctuation of horizontal temperature against vertical. Issue the following commands:

>>> plt.rcParams['figure.figsize'] = (10.0, 5.0)
>>> plt.subplot(121); plt.plot(t,Z)
>>> plt.subplot(122); plt.plot(Y,Z)
>>> plt.show()

This produces the following the plots that show vertical temperature with respect to time (left-hand side plot) and horizontal versus vertical temperature (right-hand side plot):