Linearizability of flows by embeddings
This work addresses a foundational problem in dynamical systems theory, offering new insights into linearizability with implications for symmetry, topology, and invariant manifold theory, but it appears incremental as it builds on existing theorems.
The paper tackles the problem of identifying which continuous-time dynamical systems can be globally linearized via embedding into higher-dimensional linear systems, providing necessary and sufficient conditions for such linearizing embeddings on connected state spaces with compactness properties. The results include checkable necessary conditions and extensions of classical theorems like Hartman-Grobman and Floquet normal forms.
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.