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 in living systems. One of the standard theoretical methods for analyzing synchronization dynamics of oscillatory systems is the phase reduction for weakly perturbed limit-cycle oscillators, which allows us to simplify nonlinear dynamical models exhibiting stable limit-cycle oscillations to a simple one-dimensional phase equation. Recently, the classical phase reduction method has been generalized to infinite-dimensional oscillatory systems such as spatially extended systems and time-delayed systems, and also to include amplitude degrees of freedom representing deviations of the system state from the unperturbed limit cycle. In this chapter, we discuss the method of phase-amplitude reduction for spatially extended reaction-diffusion systems exhibiting stable oscillatory patterns. As an application, we analyze entrainment of a reaction-diffusion system exhibiting limit-cycle oscillations by an optimized periodic forcing and additional feedback stabilization.