Applications of equivariant degree for gradient maps to symmetric Newtonian systems

We consider G = Γ × S 1 with Γ being a finite group, for which the complete Euler ring structure in U ( G ) is described. The multiplication tables for Γ = D 6 , S 4 and A 5 are provided in the Appendix. The equivariant degree for G -orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the G -orthogonal degree extends the degree for G -gradient maps (in the case of G = Γ × S 1 ) introduced by Gȩba in [K. Gȩba, W. Krawcewicz, J. Wu, An equivariant degree with applications to symmetric bifurcation problems I: Construction of the degree, Bull. London. Math. Soc. 69 (1994) 377–398]. The computational results obtained are applied to a Γ -symmetric autonomous Newtonian system for which we study the existence of 2 π -periodic solutions. For some concrete cases, we present the symmetric classification of the solution set for the systems considered.