Invariants and reversibility in polynomial systems of ODEs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911822327578624 |
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| author | Grašič, Mateja Jarrah, Abdul Salam Romanovski, Valery G. |
| author_facet | Grašič, Mateja Jarrah, Abdul Salam Romanovski, Valery G. |
| contents | This paper explores a relationship between invariants of certain group actions and the time-reversibility of two-dimensional polynomial differential systems exhibiting a $1:-1$ resonant singularity at the origin. We focus on the connection of time-reversibility with the Sibirsky subvariety of the center (integrability) variety, which encompasses systems possessing a local analytic first integral near the origin. An algorithm for generating the Sibirsky ideal for these systems is proposed and the algebraic properties of the ideal are examined.
Furthermore, using a generalization of the concept of time-reversibility we study $n$-dimensional systems with a $1:ζ:ζ^2:\dots:ζ^{n-1}$ resonant singularity at the origin, where $n$ is prime and $ζ$ is a primitive $n$-th root of unity. We study the invariants of a Lie group action on the parameter space of the system, leveraging the theory of binomial ideals as a fundamental tool for the analysis. Our study reveals intriguing connections between generalized reversibility, invariants, and binomial ideals, shedding light on their complex interrelations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01817 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariants and reversibility in polynomial systems of ODEs Grašič, Mateja Jarrah, Abdul Salam Romanovski, Valery G. Dynamical Systems This paper explores a relationship between invariants of certain group actions and the time-reversibility of two-dimensional polynomial differential systems exhibiting a $1:-1$ resonant singularity at the origin. We focus on the connection of time-reversibility with the Sibirsky subvariety of the center (integrability) variety, which encompasses systems possessing a local analytic first integral near the origin. An algorithm for generating the Sibirsky ideal for these systems is proposed and the algebraic properties of the ideal are examined. Furthermore, using a generalization of the concept of time-reversibility we study $n$-dimensional systems with a $1:ζ:ζ^2:\dots:ζ^{n-1}$ resonant singularity at the origin, where $n$ is prime and $ζ$ is a primitive $n$-th root of unity. We study the invariants of a Lie group action on the parameter space of the system, leveraging the theory of binomial ideals as a fundamental tool for the analysis. Our study reveals intriguing connections between generalized reversibility, invariants, and binomial ideals, shedding light on their complex interrelations. |
| title | Invariants and reversibility in polynomial systems of ODEs |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2309.01817 |