Introduction to the Modeling and Analysis of Complex Systems

160 CHAPTER 9. CHAOSreturn log(abs(1 - 2*x))def lyapunov_exponent():initialize()for t in xrange(100):update()observe()return mean(result)rvalues = arange(0, 2, 0.01)lambdas = [lyapunov_exponent() for r in rvalues]plot(rvalues, lambdas)plot([0, 2], [0, 0])xlabel(’r’)ylabel(’Lyapunov exponent’)show()Figure 9.6 shows the result. By comparing this figure with the bifurcation diagram (Fig. 8.10),you will notice that the parameter range where the Lyapunov exponent takes positive valuesnicely matches the range where the system shows chaotic behavior. Also, whenevera bifurcation occurs (e.g., r = 1, 1.5, etc.), the Lyapunov exponent touches the λ = 0 line,indicating the criticality of those parameter values. Finally, there are several locations inthe plot where the Lyapunov exponent diverges to negative infinity (they may not look so,but they are indeed going infinitely deep). Such values occur when the system convergesto an extremely stable equilibrium point with dF t /dx| x=x0 ≈ 0 for certain t. Since the definitionof the Lyapunov exponent contains logarithms of this derivative, if it becomes zero,the exponent diverges to negative infinity as well.Exercise 9.3 Plot the Lyapunov exponent of the logistic map (Eq. (8.42)) for 0 0:x t = cos 2 (rx t−1 ) (9.8)