Integrals and chaos in generalized Hénon-Heiles Hamiltonians
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arXiv
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| Format: | Preprint |
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2025
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| author | Contopoulos, G. Tzemos, A. C. Zanias, F. |
| author_facet | Contopoulos, G. Tzemos, A. C. Zanias, F. |
| contents | We study the approximate (formal) integrals of motion in the Hamiltonian $ H = \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + x^2 + y^2 \right) + ε\,\left( xy^2 + αx^3\right)$ which is an extension of the usual Hénon-Heiles Hamiltonian that has $α= -1/3$. We compare the theoretical surfaces of section (at $y=0$) with the exact surfaces of section calculated by integrating numerically many orbits. For small $ε$, the invariant curves of the theoretical and the exact surfaces of section are close to each other, but for large $ε$ there are differences. The most important is the appearance of chaos in the exact case, which becomes dominant as $ε$ approaches the escape perturbation for $α<0$. We study in particular the cases $α= 1/3$, which represents an integrable system, and $α= 0$. Finally we examine the generation of chaos through the resonance overlap mechanism in the case $α=-1/3$ (the original Hénon-Heiles system) by showing both the homoclinic and the heteroclinic intersection of the asymptotic curves of the unstable periodic orbits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_08437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrals and chaos in generalized Hénon-Heiles Hamiltonians Contopoulos, G. Tzemos, A. C. Zanias, F. Chaotic Dynamics Exactly Solvable and Integrable Systems We study the approximate (formal) integrals of motion in the Hamiltonian $ H = \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + x^2 + y^2 \right) + ε\,\left( xy^2 + αx^3\right)$ which is an extension of the usual Hénon-Heiles Hamiltonian that has $α= -1/3$. We compare the theoretical surfaces of section (at $y=0$) with the exact surfaces of section calculated by integrating numerically many orbits. For small $ε$, the invariant curves of the theoretical and the exact surfaces of section are close to each other, but for large $ε$ there are differences. The most important is the appearance of chaos in the exact case, which becomes dominant as $ε$ approaches the escape perturbation for $α<0$. We study in particular the cases $α= 1/3$, which represents an integrable system, and $α= 0$. Finally we examine the generation of chaos through the resonance overlap mechanism in the case $α=-1/3$ (the original Hénon-Heiles system) by showing both the homoclinic and the heteroclinic intersection of the asymptotic curves of the unstable periodic orbits. |
| title | Integrals and chaos in generalized Hénon-Heiles Hamiltonians |
| topic | Chaotic Dynamics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2501.08437 |