New Aspects of the Hénon–Heiles System

After summarizing the basic results for the iconic Hénon–Heiles system, we discuss two aspects: first, the characterization of chaos when moving into the chaotic regime and second the presence of long-periodic solutions between tori. We study the transition to chaos by identifying heteroclinic tangles. Considering solutions on bounded energy manifolds, we list global 1-parameter families of periodic solutions and their period-doubling bifurcations. The Poincaré–Birkhoff geometric theorem for two-dimensional area-preserving maps leads to the existence and explicit calculation of long-periodic solutions between the invariant tori. Comparison with an integrable Hamiltonian displaying the same symmetry shows the wide applicability of the geometric theorem.