Rotating spirals in oscillatory media with nonlocal interactions and their normal form

TL;DR

This paper proves the existence of rotating spiral wave solutions in oscillatory media with nonlocal interactions, using a novel combination of multiple-scales analysis and Lyapunov-Schmidt reduction to derive a normal form.

Contribution

It introduces a new analytical framework for nonlocal oscillatory media, extending the understanding of spiral wave phenomena beyond local diffusive systems.

Findings

Existence of rotating wave solutions in nonlocal media

Development of a normal form capturing leading order behavior

Application of combined analytical techniques for nonlocal systems

Abstract

Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions.