Phase and Amplitude Description of Complex Oscillatory Patterns in Reaction-Diffusion Systems

Spontaneous rhythmic oscillations are widely observed in various real-world systems. In particular, biological rhythms, which typically arise via synchronization of many self-oscillatory cells, often play important functional roles living One the standard theoretical methods for analyzing dynamics oscillatory systems is phase reduction weakly perturbed limit-cycle oscillators, allows us to simplify nonlinear dynamical models exhibiting stable a simple one-dimensional equation. Recently, classical method has been generalized infinite-dimensional such as spatially extended and time-delayed systems, also include amplitude degrees freedom representing deviations system state from unperturbed limit cycle. this chapter, we discuss phase-amplitude reaction-diffusion patterns. As an application, analyze entrainment by optimized periodic forcing additional feedback stabilization.