Applications of equivariant degree for gradient maps to symmetric Newtonian systems Original Research Article

We consider G=Γ×S1G=Γ×S1 with ΓΓ being a finite group, for which the complete Euler ring structure in U(G)U(G) is described. The multiplication tables for Γ=D6Γ=D6, S4S4 and A5A5 are provided in the Appendix. The equivariant degree for GG-orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the GG-orthogonal degree extends the degree for GG-gradient maps (in the case of G=Γ×S1G=Γ×S1) 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 ΓΓ-symmetric autonomous Newtonian system for which we study the existence of 2π2π-periodic solutions. For some concrete cases, we present the symmetric classification of the solution set for the systems considered.