A remark on omega limit sets for non-expansive dynamics
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2024
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| author | Duvall, Alon Sontag, Eduardo D. |
| author_facet | Duvall, Alon Sontag, Eduardo D. |
| contents | In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $ω$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $ω$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $ω$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are contractive with respect to any strictly convex norm (such as $\ell^p$ for any $p\not=1,\infty$), and if there is at least one bounded solution, then the $ω$-limit set of every trajectory is also an omega limit set of a linear time-invariant system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_02352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A remark on omega limit sets for non-expansive dynamics Duvall, Alon Sontag, Eduardo D. Dynamical Systems In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $ω$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $ω$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $ω$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are contractive with respect to any strictly convex norm (such as $\ell^p$ for any $p\not=1,\infty$), and if there is at least one bounded solution, then the $ω$-limit set of every trajectory is also an omega limit set of a linear time-invariant system. |
| title | A remark on omega limit sets for non-expansive dynamics |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2404.02352 |