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  "body": "\n## Summary\n\nA Lyapunov exponent measures the exponential rate at which a solution of a differential equation grows or decays as time goes to infinity.\nFor a linear system it is the upper limit of $\\frac{1}{t} \\ln \\norm{x(t)}$ along a nonzero solution $x$.\nFor a nonlinear system one applies it to the linearised equation along a trajectory, and it then measures how fast nearby trajectories separate.\nA negative exponent signals contraction, related to [[Lyapunov stability|0F2SYCKDT2]], and a positive one signals sensitive dependence on initial conditions, a mark of chaos.\nSee the [[definition|QREYWGEH41]].\n\n## Explanations\n\n### L1 human\n\nWatch how the size of a solution changes over a long time and ask for the best fitting exponential rate $e^{\\lambda t}$.\nThat rate $\\lambda$ is the Lyapunov exponent.\nFor the scalar equation $x' = a x$, the solution $x(t) = e^{a t} x(0)$ has exponent exactly $a$.\nIf two nearby starting points drift apart like $e^{\\lambda t}$ with $\\lambda \u003e 0$, small errors in the start blow up quickly.\nThis is why weather is hard to forecast far ahead.\n",
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