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

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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 Hénon–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 Hénon–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”.

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Notes

  1. TIDES, a Taylor Series Integrator for Differential EquationS. https://sourceforge.net/projects/tidesodes/.

  2. CAPD, Computer-Assisted Proofs in Dynamics: A C++ Package for Rigorous Numerics, http://capd.ii.uj.edu.pl.

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Acknowledgements

RB has been supported by the Spanish Ministry of Economy and Competitiveness (Grant PGC2018-096026-B-I00), the European Social Fund (EU) and Aragón Government (Group E24-17R), and the University of Zaragoza-CUD (grant UZCUD2019-CIE-04). DW has been supported by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453.

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  1. Departamento de Matemática Aplicada and IUMA, University of Zaragoza, Zaragoza, Spain

    Roberto Barrio

  2. Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

    Daniel Wilczak

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  1. Roberto Barrio
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Correspondence to Roberto Barrio.

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Barrio, R., Wilczak, D. Distribution of stable islands within chaotic areas in the non-hyperbolic and hyperbolic regimes in the Hénon–Heiles system. Nonlinear Dyn 102, 403–416 (2020). https://doi.org/10.1007/s11071-020-05930-x

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