Hartman-Grobman theorem

Let $𝐹 : ℝ 𝑛 \to ℝ 𝑛$ be differentiable. If $𝑥 *$ is a fixed point such that the Jacobian matrix $𝐴 = 𝐷 𝐹 (𝑥 *)$ has only eigenvalues with non-zero real parts, then there exists a neighbourhood $𝑈$ of $𝑥 *$ in which the dynamics of $˙ 𝑥 = 𝐹 (𝑥)$ is equivalent to the linear system $˙ 𝑦 = 𝐴 𝑦$ . #m/thm/dynamics

Proof

#missing/proof No proof given in @walkerMATH3021NonlinearDynamics2021

Such points are called hyperbolic fixed points. See Linear stability analysis.

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