Order and Disorder both Spatial and Temporal near a Hopf Bifurcation

Abstract

In this chapter we learn to describe the type of bifurcation from which both spatial patterns and temporal oscillations emerge (Hopf bifurcation). Using symmetry, we show that these systems are described by a universal equation (that is, independently of the system under consideration) in the neighborhood of the instability threshold. Unlike the stationary bifurcation, stationary in time, a Hopf bifurcation can lead to a dizzying array of rich behaviors, ranging from order (in the form of bands which move with time – progressive waves), to spatio-temporal chaos.