Periodic Solutions to Reversible Second Order Autonomous DDEs in Prescribed Symmetric Nonconvex Domains

TL;DR

This paper investigates the existence of symmetric periodic solutions in reversible second order delay differential equations within prescribed domains using equivariant degree theory, providing both abstract results and a concrete example.

Contribution

It introduces a novel application of Brouwer equivariant degree theory to find periodic solutions in symmetric domains for reversible delay differential equations.

Findings

Existence of symmetric periodic solutions established.

Application of equivariant degree theory to DDEs demonstrated.

Concrete example with dihedral group D8 provided.

Abstract

The existence and spatio-temporal patterns of $2\pi$-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. The solutions are supposed to take their values in a prescribed symmetric domain $D$, while $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system. The group $\Gamma$ reflects symmetries of $D$ and/or possible coupling in the corresponding network of identical oscillators, and $\mathbb Z_2$ is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of $\partial D$, Hartman-Nagumo type {\it a priori bounds} and Brouwer equivariant degree techniques, are supported by a concrete example with $\Gamma = D_8$ -- the dihedral group of order $16$.