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 $O(2) \times \Gamma \mathbb Z_2$-equivariant degree theory, where $O(2)$ is related reversing symmetry, $\Gamma$ reflects symmetric character coupling in corresponding network $\mathbb Z_2$ oddness right-hand side. Abstract results supported by a concrete example $\Gamma = D_6$ - dihedral group 12.