Solutions of Fixed Period in the Nonlinear Wave Equation on Networks

The wave equation on network is defined by ${\mathrm{\partial }}_{tt}u={\mathrm{\Delta }}_{G}u+g(u)$, where $u\in {\mathbb{R}}^n$ and the graph Laplacian ${\mathrm{\Delta }}_G$ is an operator on functions on $n$ vertices. We suppose that $g:$\mathbb{R}$^{n}\rightarrow $\mathbb{R}$^{n}$isanoddcontinuousfunctionthatsatisfies$g(0)=g^{\prime }(0)=0$ and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations $\mathrm{\Gamma }$, using a $\mathrm{\Gamma }$-equivariant topological invariant we prove the existence of multiple non-constant $p$-periodic solutions characterized by their symmetries.