One of the fundamental ways that a stationary dynamical system can become time dependent as some parameter is varied is via a Hopf bifurcation (see Appendix 1). In the supercritical case, a fixed point becomes unstable at the same time that a stable periodic orbit grows smoothly out of the fixed point. In this chapter, we use amplitude equations and comparisons of calculations with experiments to discuss the universal dynamics that arise near the onset of a Hopf bifurcation in a spatially extended homogeneous nonequilibrium medium. Although many of the concepts and issues are similar to those already discussed in Chapters 6–8 for the type I-s instability (e.g. amplitude equations, stability balloons, defects, phase equations), a new feature of oscillatory media is the appearance of propagating waves. For media with one extended direction, there are typically right- and left-propagating waves that interact in a nonlinear way with each other, and these waves can also interact nonlinearly with waves generated by reflection from a lateral boundary. An intriguing one-dimensional example that we discuss later in this chapter is the blinking state, which can be observed when a binary fluid (e.g. a mixture of water and alcohol) convects in a narrow rectangular domain, see Fig. 10.11. In a two dimensional oscillatory medium, the propagating waves most often take the form of rotating spirals (see Figures 1.9, 1.18(a), 10.3, and 10.4).

Propagating waves and spiral structures also are observed in so-called excitable media.