The Hénon–Heiles Hamiltonian near the critical energy level—some rigorous results

We continue the investigation of the Hénon–Heiles system started in (Arioli G and Zgliczyński P 2001 Symbolic dynamics for the Henon–Heiles Hamiltonian on the critical level J. Diff. Eqns 171 173–202) and we provide new results in four directions. We prove the existence of infinitely many solutions which are homoclinic or heteroclinic to periodic solutions. We prove the existence of infinitely many symmetric hyperbolic periodic solutions. We provide a new topological method to prove the existence of elliptic periodic solutions and we apply it to the system. We give all the results in an explicitly given energy interval containing the critical value of 1/6. All proofs are computer assisted.