Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition
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| Format: | Preprint |
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2026
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| _version_ | 1866910096701784064 |
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| author | Tarigo, Juan P. Stari, Cecilia Leonel, Edson D. Marti, Arturo C. |
| author_facet | Tarigo, Juan P. Stari, Cecilia Leonel, Edson D. Marti, Arturo C. |
| contents | We study the phase-space organization of the planar elastic pendulum as a function of its two dimensionless control parameters: the reduced energy $R$ and the squared frequency ratio $μ$. By randomly sampling the isoenergetic volume to classify trajectories as oscillatory, rotational, or chaotic across the $(μ, R)$ parameter plane, we obtain a global portrait of the coexistence and competition between dynamical regimes. The chaotic fraction is not uniformly distributed across the parameter plane but concentrates in a well-defined central cloud whose ridge follows a linear relation in the $(μ, R)$ plane and whose maximum does not exceed $70\%$ of the available phase space. The order-chaos-order transition is not a global property of the parameter plane but occurs specifically in the central region surrounding this cloud: along paths that traverse it, oscillatory orbits progressively give way to chaotic trajectories, which in turn yield to rotational orbits as the energy grows, revealing a clear sequential mechanism underlying the transition. The onset of rotational motion is gradual rather than sharp, reflecting a strong dependence on initial conditions. By decomposing the total energy into spring-like, pendulum-like, and coupling contributions, we establish a direct correspondence between the coupling power and the abundance of chaotic trajectories, showing that enhanced inter-mode energy exchange is a reliable indicator of dynamical complexity. These results provide a comprehensive and quantitative map of the dynamical regimes of the elastic pendulum, clarifying the structure of the chaotic cloud and connecting it to the underlying mode-coupling mechanisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_01503 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition Tarigo, Juan P. Stari, Cecilia Leonel, Edson D. Marti, Arturo C. Chaotic Dynamics We study the phase-space organization of the planar elastic pendulum as a function of its two dimensionless control parameters: the reduced energy $R$ and the squared frequency ratio $μ$. By randomly sampling the isoenergetic volume to classify trajectories as oscillatory, rotational, or chaotic across the $(μ, R)$ parameter plane, we obtain a global portrait of the coexistence and competition between dynamical regimes. The chaotic fraction is not uniformly distributed across the parameter plane but concentrates in a well-defined central cloud whose ridge follows a linear relation in the $(μ, R)$ plane and whose maximum does not exceed $70\%$ of the available phase space. The order-chaos-order transition is not a global property of the parameter plane but occurs specifically in the central region surrounding this cloud: along paths that traverse it, oscillatory orbits progressively give way to chaotic trajectories, which in turn yield to rotational orbits as the energy grows, revealing a clear sequential mechanism underlying the transition. The onset of rotational motion is gradual rather than sharp, reflecting a strong dependence on initial conditions. By decomposing the total energy into spring-like, pendulum-like, and coupling contributions, we establish a direct correspondence between the coupling power and the abundance of chaotic trajectories, showing that enhanced inter-mode energy exchange is a reliable indicator of dynamical complexity. These results provide a comprehensive and quantitative map of the dynamical regimes of the elastic pendulum, clarifying the structure of the chaotic cloud and connecting it to the underlying mode-coupling mechanisms. |
| title | Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition |
| topic | Chaotic Dynamics |
| url | https://arxiv.org/abs/2604.01503 |