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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$ and let $f : U \\to \\mathbb{R}^n$ be a function.\nThe autonomous system with vector field $f$ is the system of [[ordinary differential equations|0P97V20RW7]]\n$$x'(t) = f\\left(x(t)\\right).$$\nA [[solution|04SW7WECS8]] is a differentiable function $\\gamma : I \\to U$ on an interval $I$ with $\\gamma'(t) = f(\\gamma(t))$ for each $t \\in I$.\nThe set $U$ is the phase space, or state space, of the system.\n\n## Remarks\n\nThe right side depends on the state $x$ only, not on the time $t$ directly.\nA non-autonomous system $x' = f(t, x)$ becomes autonomous with the extra state variable $s$ and the extra equation $s' = 1$.\nThe equation $x' = k x$ of [[exponential growth|00SVP6W3CB]] and the [[logistic equation|0CC4VC7KR3]] are autonomous. The equation $x' = t x$ is not.\nSolutions of an autonomous system can be [[translated in time|0WA7ZR95V4]].\n\n## Explanations\n\n### L1 human\n\nIn an autonomous system, the law of motion does not change with time.\nThe velocity depends only on where the state is, not on when it is there.\n",
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