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