Linearizability of flows by embeddings

Abstract: 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 finite-dimensional Euclidean space. We solve this problem for dynamical systems having either a compact state space or at least one 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. Overall, our results reveal relationships between linearizability, symmetry, topology, and invariant manifold theory. In particular, these relationships yield fundamental limitations and capabilities of algorithms from the "applied Koopman operator theory" literature.