  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.
