Equivalence of differential systems

Two autonomous systems $˙ 𝑥 = 𝑓 (𝑥)$ and $˙ 𝑦 = 𝑔 (𝑦)$ with $𝑥, 𝑦 \in ℝ 𝑛$ are topologically equivalent (sometimes called conjugate) on a domains $𝑈 𝑥, 𝑈 𝑦 \in ℝ 𝑛$ iff there exists a homeomorphism $Ψ : 𝑈 𝑥 \to 𝑈 𝑦$ which preserves trajectories, i.e. if $𝑞 (𝑡, 𝑥)$ and $𝑟 (𝑡, 𝑦)$ give the trajectories of $𝑥$ and $𝑦$ respectively, then $𝑟 (𝑡, Ψ (𝑥)) = Ψ (𝑞 (𝑡, 𝑥))$ for all $𝑥 \in ℝ 𝑛$ , $𝑡 \in ℝ$ with $𝑞 (𝑡, 𝑥) \in 𝑈 𝑥$ . #m/def/dynamics/flow

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