Periodic Solutions to Reversible Second Order Autonomous Systems with Commensurate Delays

Existence and spatio-temporal patterns of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $O(2)\times \mathrm{\Gamma }\times {\mathbb{Z}}_2$-equivariant degree theory, where $O(2)$ is related to the reversing symmetry, $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }=D_6$ – the dihedral group of order 12.