Periodic Solutions to Reversible Second Order Autonomous Systems With Commensurable Delays

In this dissertation, we develop a method based on the equivariant Brouwer degree to study the existence and spatio-temporal symmetries of periodic solutions to second order reversible equivariant autonomous systems with commensurate delays. The considered delayed system of differential equations with Γ-symmetries is reformulated as a nonlinear operator equation in appropriate functional space. Then the Brouwer O(2)ΓZ2-equivariant degree theory is applied to prove the existence of 2π-periodic solutions. The group O(2) is related to the reversing symmetry, Γ reflects the spatial symmetries of the system (for instance related to symmetries of a network of coupled identical oscillators), and Z2 is related to the oddness of the right-hand-side. Abstract results are supported by a concrete example with Γ = D6 – the dihedral group of order 12.