Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914715018461184 |
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| author | Yagasaki, Kazuyuki |
| author_facet | Yagasaki, Kazuyuki |
| contents | We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense 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 real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homocliic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04803 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits Yagasaki, Kazuyuki Dynamical Systems 37J30, 34C15, 37J40, 34E10, 37C29, 37C37 We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense 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 real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homocliic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system. |
| title | Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits |
| topic | Dynamical Systems 37J30, 34C15, 37J40, 34E10, 37C29, 37C37 |
| url | https://arxiv.org/abs/2205.04803 |