Introduction to the Modeling and Analysis of Complex Systems

168 CHAPTER 9. CHAOScan’t cross, which imposes limitations on where you can go in the future. In such an increasinglyconstraining environment, it is not possible to maintain continuously exploringdynamics for an indefinitely long period of time.Exercise 9.8 Gather a pen and a piece of blank paper. Start drawing a continuouscurve on the paper that represents the trajectory of a hypothetical dynamicalsystem in a 2-D phase space. The shape of the curve you draw can be arbitrary,but with the following limitations:• You can’t let the pen go off the paper. The curve must be drawn in one continuousstroke.• The curve can’t merge into or cross itself.• You can’t draw curves flowing in opposing directions within a very tiny area(this violates the assumption of phase space continuity).Keep drawing the curve as long as you can, and see what happens. Discuss theimplications of the result for dynamical systems. Then consider what would happenif you drew the curve in a 3-D space instead of 2-D.Exercise 9.9 Let z i denote the value of the i-th peak of z(t) produced by theLorenz equations. Obtain time series data {z 1 , z 2 , z 3 , . . .} from numerical simulationresults. Plot z t against z t−1 , like in a cobweb plot, and see what kind of structureyou find there. Do this visualization for various values of r, while keeping s = 10and b = 3, and compare the results with the results of the bifurcation analysisobtained in Exercise 9.6.As reviewed through this and previous chapters, bifurcations and chaos are the mostdistinctive characteristics of nonlinear systems. They can produce unexpected systembehaviors that are often counter-intuitive to our everyday understanding of nature. Butonce you realize the possibility of such system behaviors and you know how and whenthey can occur, your view of the world becomes a more informed, enriched one. After all,bifurcations and chaos are playing important roles in our physical, biological, ecological,and technological environments (as well as inside our bodies and brains). They shouldthus deserve our appreciation.This chapter concludes our journey through systems with a small number of variables.We will shift gears and finally go into the realm of complex systems that are made of a