Long time evolution of the Hénon-Heiles system for small energy
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913585082400768 |
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| author | Costin, Ovidiu Costin, Rodica Sehgal, Kriti |
| author_facet | Costin, Ovidiu Costin, Rodica Sehgal, Kriti |
| contents | The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long time evolution of the Hénon-Heiles system for small energy Costin, Ovidiu Costin, Rodica Sehgal, Kriti Classical Analysis and ODEs Mathematical Physics Dynamical Systems The Hénon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics. In this work, we investigate the system's solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales -- far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system's inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants. The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system's long-term behavior. |
| title | Long time evolution of the Hénon-Heiles system for small energy |
| topic | Classical Analysis and ODEs Mathematical Physics Dynamical Systems |
| url | https://arxiv.org/abs/2411.16071 |