In the previous chapter generic features of spiral wave dynamics in oscillatory media were described on the basis of the complex Ginzburg–Landau equation. Spiral waves can also exist in complex oscillatory media where the local dynamics can have period-doubled or even chaotic oscillations. In regimes where complexoscillatory behavior is found, the new feature that appears in spiral waves is a line defect across which the phase of the oscillation changes by 2pi;. The presence of line defects leads to spatiotemporal patterns not seen in media with simple local oscillatory dynamics.

Complex periodic or chaotic oscillations do not have simple single-loop trajectories in concentration phase space. For example, a period-n limit cycle is described by a period-n orbit that loops n times in concentration phase space before closing on itself (see Fig. 27.1). In such circumstances no simple single-valued angle variable may be introduced to play the role of the phase. It is often possible to generalize the definition of phase, even for systems whose dynamics is chaotic, and this is related to the phenomenon of phase synchronization (Rosenblum et al., 1997; Pikovsky et al., 2001; Osipov et al., 2003).

A spiral wave is an example of a self-organized structure that is a result of phase synchronization in a medium with complex local dynamics. Reaction–diffusion equation studies (Goryachev and Kapral, 1996a, 1996b; Goryachev et al., 1998, 2000) and experiments (Yoneyama et al., 1995; Park and Lee, 1999, 2002; Guo et al., 2004; Park et al., 2004) have demonstrated that spiral waves with synchronization defect lines exist in spatially distributed systems that undergo period-doubling bifurcations.