Limit cycles of piecewise smooth vector fields on torus with non-regular switching manifold
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916895674859520 |
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| author | de Oliveira, Thaylon Souza Martins, Ricardo Miranda |
| author_facet | de Oliveira, Thaylon Souza Martins, Ricardo Miranda |
| contents | We study the maximum number of crossing limit cycles in piecewise-smooth vector fields on the two-dimensional torus, where the discontinuity set is the boundary of the fundamental square. Under the assumption of a polynomial first integral of degree n, we apply algebraic curve-intersection methods to obtain sharp upper bounds on cycle counts. In the quadratic case (n=2), we prove at most one cycle of type aa or bb and at most two of type aba or bab, and we give explicit parameter criteria for their existence. For general n, we show that cycles of type bb are bounded by n-1 and those of type aba by n(n-1). Finally, we construct concrete examples achieving these bounds, demonstrating their optimality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit cycles of piecewise smooth vector fields on torus with non-regular switching manifold de Oliveira, Thaylon Souza Martins, Ricardo Miranda Dynamical Systems We study the maximum number of crossing limit cycles in piecewise-smooth vector fields on the two-dimensional torus, where the discontinuity set is the boundary of the fundamental square. Under the assumption of a polynomial first integral of degree n, we apply algebraic curve-intersection methods to obtain sharp upper bounds on cycle counts. In the quadratic case (n=2), we prove at most one cycle of type aa or bb and at most two of type aba or bab, and we give explicit parameter criteria for their existence. For general n, we show that cycles of type bb are bounded by n-1 and those of type aba by n(n-1). Finally, we construct concrete examples achieving these bounds, demonstrating their optimality. |
| title | Limit cycles of piecewise smooth vector fields on torus with non-regular switching manifold |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2508.09306 |