Dynamics in State Space: One and Two Dimensions

Abstract

Abstract The description of nonlinear behaviour using state space techniques is introduced in this chapter by examining systems with one or two dynamical variables, that is, the state spaces have one or two dimensions. The important notions of attractors, fixed points, limit cycles, bifurcations, and stability are introduced using simple models. For systems described by differential equations, the behaviours in one and two-dimensional state spaces are quite limited due to the ‘no-intersection theorem’ for deterministic systems. Models described in this chapter are the (linear) simple harmonic oscillator and the Brusselator chemical reaction model. Taylor series expansions allow us to characterize the stability of behaviour near fixed points and limit cycles. Poincare sections are introduced as a way of reducing the effective dimensionality of systems displaying limit cycles. Bifurcation theory characterizes the sudden changes in behaviour that can occur for nonlinear systems as their parameters are changed.