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  "body": "\n## Summary\n\nA Lyapunov function for a difference equation $x_{n+1} = f(x_n)$ is a function that is positive away from an [[equilibrium|D62PEKHRV5]] and does not increase along solutions.\nIt is the discrete-time version of the [[Lyapunov function|01REYCJ371]] of a differential equation.\nIts existence proves [[stability|CHEQ235HW9]] of the equilibrium without solving the equation, and a strict decrease proves asymptotic stability.\nThis is the main method for nonlinear difference equations whose linearization gives no answer.\n\n## Explanations\n\n### L1 human\n\nThink of the function as an energy or a height above the bottom of a bowl.\nAt each step of the recursion the energy goes down or stays the same, so the state cannot climb away from the bottom.\nFor $x_{n+1} = x_n / 2$ the square $V(x) = x^2$ works: it changes to $x^2/4$ at each step, which is smaller.\nSo every solution slides down to $0$.\n",
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