Bifurcations and Chaos in Dynamical Systems

Complex systems theory deals with dynamical systems containing often large numbers of variables. It extends dynamical systems theory, which treats dynamical systems containing a few variables. A good understanding of dynamical systems theory is therefore a prerequisite when studying complex systems. In this chapter we introduce core concepts, like attractors and Lyapunov exponents, bifurcations, and deterministic chaos from the realm of dynamical systems theory. An introduction to catastrophe theory will be provided together with the notion of rate-induced tipping and colliding attractors. Most of the chapter will be devoted to ordinary differential equations and maps, the traditional focus of dynamical systems theory, venturing however towards the end into the intricacies of time delay dynamical systems.