Introduction to the Modeling and Analysis of Complex Systems

162 CHAPTER 9. CHAOSThen explain in words how its dynamics change over r.9.4 Chaos in Continuous-Time ModelsAs we reviewed above, chaos is really easy to show in discrete-time models. But thediscovery of chaos was originally made with continuous-time dynamical systems, i.e., differentialequations. Edward Lorenz, an American mathematician and meteorologist, andone of the founders of chaos theory, accidentally found chaotic behavior in the followingmodel (called the Lorenz equations) that he developed to study the dynamics of atmosphericconvection in the early 1960s [5]:dx= s(y − x)dt(9.9)dy= rx − y − xzdt(9.10)dz= xy − bzdt(9.11)Here s, r, and b are positive parameters.This model is known to be one of the first thatcan show chaos in continuous time. Let’s simulate this model with s = 10, r = 30, andb = 3, for example:Code 9.2: Lorenz-equations.pyfrom pylab import *from mpl_toolkits.mplot3d import Axes3Ds = 10.r = 30.b = 3.Dt = 0.01def initialize():global x, xresult, y, yresult, z, zresult, t, timestepsx = y = z = 1.xresult = [x]yresult = [y]zresult = [z]