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

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 $16$.

arXiv admin note: substantial text overlap with arXiv:2007.09166