Fundamental theorem of flows

Let $𝐹 : ℝ 𝑛, ℝ 𝑛$ be differentiable. Let $𝑥 0 \in ℝ 𝑛$ , $𝑈$ be a neighbourhood of $𝑥 0$ , $𝐹 (𝑥) \neq 0$ for $𝑥 \in 𝑈$ , and $𝑣 \in ℝ 𝑛$ , $𝑣 \neq 0$ . Then there exists a differentiable change of coördinates $𝑦 = Ψ (𝑥)$ such that $˙ 𝑥 = 𝐹 (𝑥)$ is equivalent to $˙ 𝑦 = 𝑣$ for $𝑥 \in 𝑈$ .¹ #m/thm/dynamics/flow

Proof

#missing/proof No proof provided in @walkerMATH3021NonlinearDynamics2021.

That is to say flows in a neighbourhood of regular points are equivalent to parallel trajectories at constant velocity; the only interesting behaviour occurs in neighbourhoods of fixed points.

#state/tidy | #lang/en | #SemBr

2021. MATH3021: Nonlinear dynamics & chaos, p. 27 ↩

Fundamental theorem of flows

Footnotes