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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be continuously differentiable, and let $x^* \\in U$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.\nThe point $x^*$ is hyperbolic if no eigenvalue of the Jacobian matrix $Df(x^*)$ has real part $0$.\n\n## Remarks\n\nA saddle, a node and a spiral of a planar linear system are hyperbolic. A centre is not.\nFor the pendulum $x' = y$, $y' = -\\sin x$, the equilibrium $(\\pi, 0)$ is hyperbolic, with eigenvalues $\\pm 1$. The equilibrium $(0, 0)$ is not, with eigenvalues $\\pm i$.\nNear a hyperbolic equilibrium, the [[linearisation|09ZWYRQ41J]] describes the system qualitatively, by the [[Hartman–Grobman theorem|0S8WX8A4V9]].\nIf all eigenvalues have negative real part, the point is a sink. If all have positive real part, it is a source. Otherwise it is a saddle.\n\n## Explanations\n\n### L1 human\n\nAt a hyperbolic equilibrium, each direction of the linearisation either contracts or expands.\nNo direction is neutral, so small changes of the field do not change the type of the equilibrium.\n",
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