Linearizability of flows by embeddings
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918431274565632 |
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| author | Kvalheim, Matthew D. Arathoon, Philip |
| author_facet | Kvalheim, Matthew D. Arathoon, Philip |
| contents | We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem for dynamical systems on connected state spaces that are either compact or contain at least one nonempty compact attractor, obtaining necessary and sufficient conditions for the existence of linearizing $C^k$ embeddings for $k\in \mathbb{N}_{\geq 0}\cup \{\infty\}$. Corollaries include (i) several checkable necessary conditions for global linearizability and (ii) extensions of the Hartman-Grobman and Floquet normal form theorems beyond the classical settings. Our results open new perspectives on linearizability by establishing relationships to symmetry, topology, and invariant manifold theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18288 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linearizability of flows by embeddings Kvalheim, Matthew D. Arathoon, Philip Dynamical Systems Systems and Control Optimization and Control 37C15, 37C79, 37C81, 37C70 We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem for dynamical systems on connected state spaces that are either compact or contain at least one nonempty compact attractor, obtaining necessary and sufficient conditions for the existence of linearizing $C^k$ embeddings for $k\in \mathbb{N}_{\geq 0}\cup \{\infty\}$. Corollaries include (i) several checkable necessary conditions for global linearizability and (ii) extensions of the Hartman-Grobman and Floquet normal form theorems beyond the classical settings. Our results open new perspectives on linearizability by establishing relationships to symmetry, topology, and invariant manifold theory. |
| title | Linearizability of flows by embeddings |
| topic | Dynamical Systems Systems and Control Optimization and Control 37C15, 37C79, 37C81, 37C70 |
| url | https://arxiv.org/abs/2305.18288 |