Solutions of Fixed Period in the Nonlinear Wave Equation on Networks

The wave equation on network is defined by ∂_{tt}u=Δ_{G}u+g(u), where u∊ℝⁿ and the graph Laplacian Δ_G is an operator on functions on n vertices. We suppose that g:ℝⁿ→ℝⁿ is an odd continuous function that satisfies g(0)=0, Dg(0)=0 and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations Γ, using a Γ-equivariant topological invariant we prove the existence of multiple non-constant p-periodic solutions characterized by their symmetries.