Sharkovskii theorem for infinite dimensional dynamical systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929705302622208 |
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| author | Gierzkiewicz, Anna Szczelina, Robert |
| author_facet | Gierzkiewicz, Anna Szczelina, Robert |
| contents | We present an adaptation of a relatively simple topological argument to show the existence of many periodic orbits in an infinite dimensional dynamical system, provided that the system is close to a one-dimensional map in a certain sense. Namely, we prove a Sharkovskii-type theorem: if the system has a periodic orbit of basic period $m$, then it must have all periodic orbits of periods $n \triangleright m$, for $n$ preceding $m$ in Sharkovskii ordering. The assumptions of the theorem can be verified with computer assistance, and we demonstrate the application of such an argument in the case of Delay Differential Equations (DDEs): we consider the Rössler ODE system perturbed by a delayed term and we show that it retains periodic orbits of all natural periods for fixed values of parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharkovskii theorem for infinite dimensional dynamical systems Gierzkiewicz, Anna Szczelina, Robert Dynamical Systems 34K13, 34K23, 34K27, 37B10, 65G20 G.1.7 We present an adaptation of a relatively simple topological argument to show the existence of many periodic orbits in an infinite dimensional dynamical system, provided that the system is close to a one-dimensional map in a certain sense. Namely, we prove a Sharkovskii-type theorem: if the system has a periodic orbit of basic period $m$, then it must have all periodic orbits of periods $n \triangleright m$, for $n$ preceding $m$ in Sharkovskii ordering. The assumptions of the theorem can be verified with computer assistance, and we demonstrate the application of such an argument in the case of Delay Differential Equations (DDEs): we consider the Rössler ODE system perturbed by a delayed term and we show that it retains periodic orbits of all natural periods for fixed values of parameters. |
| title | Sharkovskii theorem for infinite dimensional dynamical systems |
| topic | Dynamical Systems 34K13, 34K23, 34K27, 37B10, 65G20 G.1.7 |
| url | https://arxiv.org/abs/2411.19190 |