Periodic Solutions to Reversible Second Order Autonomous Systems with Commensurate Delays

TL;DR

This paper investigates the existence and patterns of periodic solutions in reversible second order autonomous systems with delays, using equivariant degree theory, and provides a concrete example involving dihedral symmetry.

Contribution

It introduces a novel application of equivariant degree theory to analyze periodic solutions in delayed reversible systems with symmetry.

Findings

Existence of periodic solutions established using equivariant degree theory.

Identification of spatio-temporal patterns related to system symmetries.

Concrete example with dihedral group D6 illustrating theoretical results.

Abstract

Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2) \times \Gamma \times \mathbb Z_2$-equivariant degree theory, where $O(2)$ is related to the reversing symmetry, $\Gamma$ reflects the symmetric character of the coupling in the corresponding network and $\mathbb Z_2$ is related to the oddness of the right-hand-side. Abstract results are supported by a concrete example with $\Gamma = D_6$ -- the dihedral group of order 12.