Abstract
This chapter introduces the reader to some of the main conceptual ideas behind dynamical systems theory from the perspective of an experimentalist. It first considers the qualitative approaches used to study complex problems before discussing dynamical systems and bifurcations. In particular, it examines the use of time series to represent solutions and dynamics in the phase space, phase space respresentations of equilibrium and nonequilibrium steady states, the qualitative analysis of steady states, and some of the mechanics of local stability analysis for an equilibrium using the Lotka–Volterra model for an equilibrium steady state. It also explores the relationship between the type of model dynamics and the geometry of the underlying mathematical functions. Finally, it presents an empirical example from ecology, Hopf bifurcation in an aquatic microcosm, to illustrate the main concepts of dynamical systems theory and shows that the mathematics of dynamical systems underlies the dynamics of real ecological systems.
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