  Systems of dynamical elements exhibiting spontaneous rhythms are found in
various fields of science and engineering, including physics, chemistry,
biology, physiology, and mechanical and electrical engineering. Such dynamical
elements are often modeled as nonlinear limit-cycle oscillators. In this
article, we briefly review phase reduction theory, which is a simple and
powerful method for analyzing the synchronization properties of limit-cycle
oscillators exhibiting rhythmic dynamics. Through phase reduction theory, we
can systematically simplify the nonlinear multi-dimensional differential
equations describing a limit-cycle oscillator to a one-dimensional phase
equation, which is much easier to analyze. Classical applications of this
theory, i.e., the phase locking of an oscillator to a periodic external forcing
and the mutual synchronization of interacting oscillators, are explained.
Further, more recent applications of this theory to the synchronization of
non-interacting oscillators induced by common noise and the dynamics of coupled
oscillators on complex networks are discussed. We also comment on some recent
advances in phase reduction theory for noise-driven oscillators and rhythmic
spatiotemporal patterns.
