Distribution of stable islands within chaotic areas in the non-hyperbolic and hyperbolic regimes in the Hénon–Heiles system

Abstract

We provide rigorous computer-assisted proofs of the existence of different dynamical objects, like stable families of periodic orbits, bifurcations and stable invariant tori around them, in the paradigmatic Henon–Heiles system. There are in the literature a large number of articles with numerical simulations on this system, and other open Hamiltonians, but only a few give a rigorous guarantee of simulations. In this article, we present the necessary link between the numerical simulations and the mathematical structure of the system, since it is relevant to provide evidence of the existence of some of the different objects detected numerically to evaluate the quality of the numerical results. Remarkably, we present a proof of the existence of stable regions in the non-hyperbolic and hyperbolic regimes classically established for the Henon–Heiles system. In particular, we prove the important results of the existence of bounded stable regular regions located within the escape region, far from the regime of the KAM islands, which are called “safe regions”.