Phase reduction analysis of traveling breathers in reaction--diffusion systems

TL;DR

This paper develops a phase reduction framework for analyzing traveling breathers in reaction-diffusion systems with spatial symmetry, revealing coupled spatiotemporal dynamics and validating results with numerical simulations.

Contribution

It introduces a novel phase reduction method for traveling breathers in reaction-diffusion systems, capturing coupled spatial and temporal phase dynamics.

Findings

Derived phase equations show nontrivial coupled spatiotemporal dynamics.

Phase differences evolve consistently with numerical simulations.

Applicable to FitzHugh--Nagumo and Gray--Scott models.

Abstract

We formulate a theory for phase reduction analysis of traveling breathers in reaction--diffusion systems with spatial translational symmetry. In this formulation, the spatial and temporal phases represent the position and oscillation of a traveling breather, respectively. We perform phase reduction analysis on a pair of FitzHugh--Nagumo models exhibiting standing breathers and a pair of Gray--Scott models exhibiting traveling breathers. The derived phase equations for the spatial and temporal phases indicate nontrivial spatiotemporal dynamics, where both phases are mutually coupled. Using the phase equations, we obtain the time evolution of the phase differences, which is consistent with that obtained from direct numerical simulations.