Periodic Solutions, Stability and Non-Integrability in a Generalized Hénon-Heiles Hamiltonian System

DOI

10.1080/14029251.2013.805567How to use a DOI?

Keywords

Generalized Hénon-Heiles Hamiltonian; periodic orbits; integrability; averaging theory

Abstract

We consider the Hamiltonian function defined by the cubic polynomial H=12(px2+py2)+12(x2+y2)+A3x3+Bxy2+Dx2y , where A, B, D ∈ ℝ are parameters and so H is an extension of the well known Hénon-Heiles problem. Our main contribution for D ≠ 0, A + B ≠ 0 and other technical restrictions are in three aspects: existence of periodic solutions, stability and instability of these periodic solutions and the problem of non-integrability of the system associated to H. Initially we give sufficient conditions on the three parameters of these generalized Hénon-Heiles systems, which guarantees that at any positive energy level, the Hamiltonian system has periodic orbits. After that, we prove that its stability changes with the values of the parameters. Finally, we show that the generalized Hénon–Heiles systems cannot have any second first integral of class 𝒞¹ in the sense of Liouville–Arnol'd. In fact, the parameters where our problem is not integrable in the sense of Liouville–Arnol'd are the same where the periodic orbits were analytically found through averaging theory.

Copyright

© 2013 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Dante Carrasco
AU  - Claudio Vidal
PY  - 2021
DA  - 2021/01/06
TI  - Periodic Solutions, Stability and Non-Integrability in a Generalized Hénon-Heiles Hamiltonian System
JO  - Journal of Nonlinear Mathematical Physics
SP  - 199
EP  - 213
VL  - 20
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2013.805567
DO  - 10.1080/14029251.2013.805567
ID  - Carrasco2021
ER  -