Introduction to the Modeling and Analysis of Complex Systems

Next phase space9.3. LYAPUNOV EXPONENT 157FoldingCurrent phase spaceFigure 9.4: Schematic illustration of stretching and folding taking place in a chaoticiterative map. The current phase space (right) is first stretched (middle right) and thenfolded (middle left, left) to generate the next phase space in each iteration.If you know binary notations of real numbers, it should be obvious that this iterative map issimply shifting the bit string in x always to the left, while forgetting the bits that came beforethe decimal point. And yet, such a simple arithmetic operation can still create chaos, ifthe initial condition is an irrational number (Fig. 9.5)! This is because an irrational numbercontains an infinite length of digits, and chaos continuously digs them out to produce afluctuating behavior at a visible scale.Exercise 9.2 The saw map can also show chaos even from a rational initial condition,if its behavior is manually simulated by hand on a cobweb plot. Explainwhy.9.3 Lyapunov ExponentFinally, I would like to introduce one useful analytical metric that can help characterizechaos. It is called the Lyapunov exponent, which measures how quickly an infinitesimally