Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908344261804032 |
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| author | Yagasaki, Kazuyuki |
| author_facet | Yagasaki, Kazuyuki |
| contents | We consider time-periodic perturbations of analytically integrable systems in the sense of Bogoyavlenskij and study their \emph{real-meromorphic} nonintegrability, using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not \emph{real-meromorphically} integrable near homo- and heteroclinic orbits. We illustrate the theory for rotational motions of a periodically forced rigid body, which provides a mathematical model of a quadrotor helicopter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits Yagasaki, Kazuyuki Dynamical Systems 37J30, 34D10, 70K44, 37C29, 34C37, 70E20 We consider time-periodic perturbations of analytically integrable systems in the sense of Bogoyavlenskij and study their \emph{real-meromorphic} nonintegrability, using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not \emph{real-meromorphically} integrable near homo- and heteroclinic orbits. We illustrate the theory for rotational motions of a periodically forced rigid body, which provides a mathematical model of a quadrotor helicopter. |
| title | Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits |
| topic | Dynamical Systems 37J30, 34D10, 70K44, 37C29, 34C37, 70E20 |
| url | https://arxiv.org/abs/2505.00088 |