Bifurcações em Redes Hamiltonianas Acopladas e o Problema dos Três Corpos - Melo, A. (Trends in Computational and Applied Mathematics, 2024)
The coupled cell formalism is a systematic way of representing and studying nonlinear coupled differential equations using directed graphs. We observed that only bidirectionally coupled digraphs can represent Hamiltonian systems. We present recent results in networks of linearized coupled Hamiltonian systems with a discussion of the Hamiltonian Hopf theorem in this context. We show that the eigenspace in a bifurcation of codimension one of a synchronous equilibrium of a regular Hamiltonian network can be expressed in terms of the eigenspaces of the adjacency matrix of the associated digraph. We display a version of the Lyapunov center theorem for this type of network.