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  "body": "\n## Summary\n\nAn [[equilibrium|D62PEKHRV5]] of a difference equation is stable if solutions that start close to it stay close to it at all later times.\nIt is asymptotically stable if, in addition, nearby solutions converge to it.\nThese notions are the discrete-time versions of [[Lyapunov stability|0F2SYCKDT2]] for differential equations.\nThey decide which equilibria can be observed in practice, for example which population levels persist in a model.\n\n## Explanations\n\n### L1 human\n\nPut a ball at the bottom of a bowl and nudge it: it stays near the bottom, so the bottom is stable.\nBalance it on top of a hill and nudge it: it rolls away, so the top is unstable.\nFor the rule $x_{n+1} = x_n / 2$, the equilibrium $0$ is asymptotically stable: every solution halves at each step and tends to $0$.\nFor $x_{n+1} = 2 x_n$, the equilibrium $0$ is unstable: every nonzero start doubles away.\n",
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