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  "body": "\n## Statement\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be continuously differentiable. Let $\\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \\in U$ be a [[hyperbolic equilibrium point|0C5PREPZWG]].\nLet $A = Df(x^*)$.\nThen there are open neighbourhoods $V \\subseteq U$ of $x^*$ and $W \\subseteq \\mathbb{R}^n$ of $0$, and a homeomorphism $h : V \\to W$ with $h(x^*) = 0$, as follows.\nFor each $x \\in V$ and each $t \\in \\mathbb{R}$ such that $\\varphi_s(x) \\in V$ for all $s$ between $0$ and $t$, $h(\\varphi_t(x)) = e^{tA} h(x)$.\n\n## Remarks\n\nSo near $x^*$ the flow is topologically conjugate to the flow of the [[linearisation|09ZWYRQ41J]] $y' = A y$, given by the [[matrix exponential|0YC8JT513J]]. The map $h$ sends trajectories to trajectories and keeps the direction of time.\nThe map $h$ is in general only continuous, not differentiable.\nThe hypothesis of hyperbolicity is necessary. For $x' = -x^3$ and $x' = x^3$ on $\\mathbb{R}$, the linearisation at $0$ is the same, but the two phase portraits are not conjugate.\n\n## History\n\nThe theorem is named after Philip Hartman and David M. Grobman. It is also called the linearization theorem.\n\n## Explanations\n\n### L1 human\n\nNear a hyperbolic equilibrium, a nonlinear system looks like its linear approximation, after a continuous change of coordinates.\nSo the pattern of trajectories of the linear system, such as a saddle or a spiral, also appears in the nonlinear system.\n",
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    "name": "Hartman–Grobman theorem",
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  "name": "Hartman–Grobman theorem",
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