Introduction to the Modeling and Analysis of Complex Systems

Chapter 9Chaos9.1 Chaos in Discrete-Time ModelsFigure 8.10 showed a cascade of period-doubling bifurcations, with the intervals betweenconsecutive bifurcation thresholds getting shorter and shorter geometrically as rincreased. This cascade of period doubling eventually leads to the divergence of the periodto infinity at r ≈ 1.7 in this case, which indicates the onset of chaos. In this mysteriousparameter regime, the system loses any finite-length periodicity, and its behavior looks essentiallyrandom. Figure 9.1 shows an example of such chaotic behavior of Eq. (8.37) withr = 1.8.So what is chaos anyway? It can be described in a number of different ways, asfollows:Chaos—• is a long-term behavior of a nonlinear dynamical system that never falls in anystatic or periodic trajectories.• looks like a random fluctuation, but still occurs in completely deterministic, simpledynamical systems.• exhibits sensitivity to initial conditions.• occurs when the period of the trajectory of the system’s state diverges to infinity.• occurs when no periodic trajectories are stable.153