Global Bifurcation of Periodic Solutions in Symmetric Reversible Second Order Systems with Delays

Abstract

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays is studied using the Brouwer O(2)×Γ×Z2-equivariant degree theory. Here, O(2) is related to the reversal symmetry combined with the autonomous form of the system, Z2 is related to the oddness of the right-hand-side. Γreflects other spatial symmetries of the system (in particular it may be related to the symmetric character of the coupling in the corresponding network). Global bifurcation results were obtained for a general reversible second order autonomous system with multiple delays which were illustrated via concrete example with Γ = D6– the dihedral group of order 12.