Three-Dimensional State Space and Chaos
This chapter extends state space descriptions to systems with three (or more) dimensions. For systems described by differential equations, higher state space dimensions allow for more complex behaviour including quasi-periodic behaviour and chaotic behaviour. Chaotic behaviour is characterized by strange attractors in state space. Intermittency and crises as new types of bifurcations are described. Stable manifolds, unstable manifolds, and homoclinic and heteroclinic orbits are used to describe three-dimensional state space behaviour. These ideas are used to understand some of the dynamics of the famous Lorenz model and the Smale horseshoe map. Lyapunov exponents provide a quantitative measure of the degree of chaos.
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