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  "body": "\n## Summary\n\nAn equilibrium is exponentially stable when nearby solutions approach it at least as fast as a decaying exponential.\nIt is a strong form of [[asymptotic stability|0NB689782W]] that comes with an explicit rate.\nFor a linear system $x' = Ax$, asymptotic and exponential stability coincide, and both mean that every eigenvalue of $A$ has negative real part.\nExponential stability survives small perturbations of the system, which makes it the preferred notion in control theory.\nSee the [[definition|V6E2R7JNN8]].\n\n## Explanations\n\n### L1 human\n\nA solution that is disturbed from rest returns to rest, and the size of the disturbance shrinks by a fixed factor in each unit of time.\nFor $x' = -x$, the solution $x_0 e^{-t}$ loses about two thirds of its size every unit of time.\nFor $x' = -x^3$, solutions also tend to $0$, but only like $1/\\sqrt{2t}$, which is slower than any exponential.\nSo the origin of $x' = -x^3$ is asymptotically stable but not exponentially stable.\n",
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