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  "body": "\n## Summary\n\nA discrete-time dynamical system is a [[dynamical system|BBZP1R872N]] in which time advances in whole steps.\nIt is given by a single map $f$ of a space to itself, and the state after $n$ steps is the [[iterate|AYVTR2ES5B]] $f^n(x)$.\nSuch systems arise from sampling a continuous motion at regular times, from population models with generations, and from numerical methods.\nMany basic phenomena of dynamics, such as chaos and period doubling, are already visible in maps of an interval.\n\n## Explanations\n\n### L1 human\n\nImagine pressing a button that changes the state of a system by a fixed rule, and pressing it again and again.\nThe sequence of states $x, f(x), f(f(x)), \\dots$ is the orbit of the starting state.\nFor example, take the rule $f(x) = x^2$ on the real numbers and start at $x = 1/2$.\nThe orbit is $1/2, 1/4, 1/16, 1/256, \\dots$, and it rushes to $0$.\nStarting at $x = 2$ instead gives $2, 4, 16, 256, \\dots$, which runs off to infinity.\n",
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