Rotating Wave Solutions to Lattice Dynamical Systems II: Persistence Results

TL;DR

This paper proves that rotating wave solutions in lattice dynamical systems persist under small coupling, using advanced functional analysis, and extends understanding beyond traditional reaction-diffusion models.

Contribution

It demonstrates the persistence of rotating wave solutions in Lambda-Omega equations for small coupling, advancing mathematical understanding of nonlinear wave dynamics.

Findings

Existence of persistent rotating wave solutions for small coupling values

Application of a non-standard Implicit Function Theorem

Extension beyond reaction-diffusion equations to more general differential systems

Abstract

This work comes as the second part in a series of investigations into the dynamics of rotating waves as solutions to lattice dynamical systems. Such nonlinear waves as solutions to mathematical equations are of great interest throughout the physical sciences due to their association with many electrophysiological pathologies and this investigation aims to further the understanding of rotating waves from a mathematical perspective. Here we focus on so-called Lambda-Omega differential equations, a well-studied generalization of the celebrated Ginzburg-Landau equation, to show that there exists an interval of sufficiently small coupling values for which a rotating wave solution persists. This result is achieved using a wide range of functional analytic tools, primarily in an effort to apply a non-standard Implicit Function Theorem. This work initiates subsequent studies into the dynamics and bifurcations of rotating waves away from the reaction-diffusion equation setting to differential equations for which traditional symmetry based centre manifold reductions cannot be applied.