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

The existence of periodic solutions to second order differential systems is a classical problem that has been studied by many authors using different methods and techniques. In this Thesis, the existence and spatio-temporal patterns of 2π-periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer O(2) × Γ × Z2-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 Γ reflects symmetries of D and/or possible coupling in the corresponding network of identical oscillaltors, and Z2 is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of ∂D, Hartman-Nagumo type a priori bounds and Brouwer equivariant degree techniques, are supported by a concrete example with Γ = D8 – the dihedral group of order 16.