# Phase reduction approach to synchronization of nonlinear oscillators

## TL;DR

This paper reviews phase reduction theory, a method for simplifying and analyzing the synchronization behavior of nonlinear limit-cycle oscillators across various scientific fields.

## Contribution

It provides a comprehensive overview of phase reduction theory and its recent extensions to complex networks, noise-driven oscillators, and spatiotemporal patterns.

## Key findings

- Simplifies multi-dimensional oscillator equations to one-dimensional phase equations.
- Explains classical and recent applications in synchronization phenomena.
- Discusses advances in phase reduction for complex and noisy systems.

## Abstract

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.