Stable Periodic Solutions to Lambda-Omega Lattice Dynamical Systems

TL;DR

This paper investigates the stability of periodic solutions, especially rotating waves, in Lambda-Omega lattice dynamical systems by transforming the problem into an infinite-dimensional fast-slow differential equation and analyzing the properties of the slow manifold.

Contribution

It introduces a novel approach to analyze stability using an invariant slow manifold and provides conditions for convergence, extending understanding of rotating wave stability in Lambda-Omega systems.

Findings

Existence of an invariant slow manifold near periodic solutions.

Conditions for algebraic convergence on the slow manifold.

Stability analysis of rotating wave solutions.

Abstract

In this manuscript we consider the stability of periodic solutions to Lambda-Omega lattice dynamical systems. In particular, we show that an appropriate ansatz casts the lattice dynamical system as an infinite-dimensional fast-slow differential equation. In a neighbourhood of the periodic solution an invariant slow manifold is proven to exist, and that this slow manifold is uniformly exponentially attracting. The dynamics of solutions on the slow manifold become significantly more complicated and require a more delicate treatment. We present sufficient conditions to guarantee convergence on the slow manifold which is algebraic, as opposed to exponential, in the slow-time variable. Of particular interest to our work in this manuscript is the stability of a rotating wave solution, recently found to exist in the Lambda-Omega systems studied herein.