Bifurcations and Chaos in Dynamical Systems

Abstract

Complex system theory deals with dynamical systems containing often a large number of variables. It extends dynamical system theory, which deals with 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 important concepts, like regular and irregular behavior, attractors and Lyapunov exponents, bifurcations, and deterministic chaos from the realm of dynamical system theory. An introduction to catastrophe theory and to the notion of global bifurcations is also provided. Most of the chapter will be devoted to ordinary differential equations and maps, the traditional focus of dynamical system theory, venturing however towards the end into the intricacies of time-delayed dynamical systems.