On the Concept of Dynamical Reduction : The Case of Coupled Oscillators

TL;DR

This paper reviews and unifies two key methods for simplifying coupled oscillators' dynamics, highlighting their structural similarities and introducing a new perturbative expansion for phase reduction in oscillator populations.

Contribution

It presents a more unified presentation of center-manifold and phase reduction theories, emphasizing their common principles and introducing a new perturbative approach for discrete oscillator populations.

Findings

Unified framework for reduction methods

Structural similarity between theories highlighted

New perturbative expansion for phase reduction

Abstract

An overview is given on two representative methods of dynamical reduction known as center-manifold reduction and phase reduction. These theories are presented in a somewhat more unified fashion than the theories in the past. The target systems of reduction are coupled limit-cycle oscillators. Particular emphasis is placed on the remarkable structural similarity existing between these theories. While the two basic principles, i.e. (i) reduction of dynamical degrees of freedom and (ii) transformation of reduced evolution equation to a canonical form, are shared commonly by reduction methods in general, it is shown how these principles are incorporated into the above two reduction theories in a coherent manner. Regarding the phase reduction, a new formulation of perturbative expansion is presented for discrete populations of oscillators. The style of description is intended to be so informal that one may digest, without being bothered with technicalities, what has been done after all under the word reduction.