Introduction to the Modeling and Analysis of Complex Systems

158 CHAPTER 9. CHAOS1.00.80.60.40.20.00.0 0.2 0.4 0.6 0.8 1.0Figure 9.5: Cobweb plot of the saw map with π/4 as the initial condition.small distance between two initially close states grows over time:∣ F t (x 0 + ε) − F t (x 0 ) ∣ ∣ ≈ εeλt(9.2)The left hand side is the distance between two initially close states after t steps, and theright hand side is the assumption that the distance grows exponentially over time. Theexponent λ measured for a long period of time (ideally t → ∞) is the Lyapunov exponent.If λ > 0, small distances grow indefinitely over time, which means the stretchingmechanism is in effect. Or if λ < 0, small distances don’t grow indefinitely, i.e., the systemsettles down into a periodic trajectory eventually. Note that the Lyapunov exponentcharacterizes only stretching, but as we discussed before, stretching is not the only mechanismof chaos. You should keep in mind that the folding mechanism is not captured inthis Lyapunov exponent.We can do a little bit of mathematical derivation to transform Eq. (9.2) into a more