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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^k\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> embeddings for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\in \mathbb {N}_{\ge 0}\cup \{\infty \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mrow> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>∪</mo> <mrow> <mo stretchy="false">{</mo> <mi>∞</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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Linearizability of flows by embeddings

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月7日

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  • Matthew D. Kvalheim,
  • Philip Arathoon

摘要

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 \(C^k\) C k embeddings for \(k\in \mathbb {N}_{\ge 0}\cup \{\infty \}\) k ∈ N ≥ 0 ∪ { ∞ } . 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.