Koopman Embedding and Super-Linearization Counterexamples with Isolated Equilibria

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2306.15126.pdf (674.67 KB)

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https://arxiv.org/abs/2306.15126

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https://doi.org/10.48550/arXiv.2306.15126
 http://hdl.handle.net/11603/30368

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UMBC Mathematics and Statistics Department

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https://orcid.org/0000-0002-2662-6760

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2023-07-19

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journal articles
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Abstract

A frequently repeated claim in the "applied Koopman operator theory" literature is that a dynamical system with multiple isolated equilibria cannot be linearized in the sense of admitting a smooth embedding as an invariant submanifold of a linear dynamical system. This claim is sometimes made only for the class of super-linearizations, which additionally require that the embedding "contain the state". We show that both versions of this claim are false by constructing (super-)linearizable smooth dynamical systems on ℝᵏ having any countable (finite) number of isolated equilibria for each k > 1.