Introduction to the Modeling and Analysis of Complex Systems

156 CHAPTER 9. CHAOS0.8Logistic map1.51.0Cubic map64Sinusoid map1.00.8Saw map0.60.520.60.40.0-0.50-20.40.2-1.0-40.20.0-1.5-60.00.0 0.2 0.4 0.6 0.8 1.0-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-6 -4 -2 0 2 4 60.0 0.2 0.4 0.6 0.8 1.0Figure 9.3: Simple maps that show chaotic behavior (for Exercise 9.1).9.2 Characteristics of ChaosIt is helpful to realize that there are two dynamical processes always going on in anykind of chaotic systems: stretching and folding [33]. Any chaotic system has a dynamicalmechanism to stretch, and then fold, its phase space, like kneading pastry dough(Fig. 9.4). Imagine that you are keeping track of the location of a specific grain of flourin the dough while a pastry chef kneads the dough for a long period of time. Stretchingthe dough magnifies the tiny differences in position at microscopic scales to a larger, visibleone, while folding the dough always keeps its extension within a finite, confined size.Note that folding is the primary source of nonlinearity that makes long-term predictions sohard—if the chef were simply stretching the dough all the time (which would look more likemaking ramen), you would still have a pretty good idea about where your favorite grain offlour would be after the stretching was completed.This stretching-and-folding view allows us to make another interpretation of chaos:Chaos can be understood as a dynamical process in which microscopic informationhidden in the details of a system’s state is dug out and expanded to a macroscopicallyvisible scale (stretching), while the macroscopic information visible in the currentsystem’s state is continuously discarded (folding).This kind of information flow-based explanation of chaos is quite helpful in understandingthe essence of chaos from a multiscale perspective. This is particularly clear when youconsider the saw map discussed in the previous exercise:x t = fractional part of 2x t−1 (9.1)