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  "body": "\n## Summary\n\nLet $f$ be a function on an interval of $\\mathbb{R}$. The phase line of the scalar [[autonomous equation|07BNRFQ0PJ]] $x' = f(x)$ is a diagram of the line of states.\nIt marks the [[equilibrium points|0G4X36AE0X]], which are the zeros of $f$. It puts an arrow on each interval between them: up where $f \u003e 0$ and down where $f \u003c 0$.\nThe arrows show the direction in which each solution moves.\n\n## Remarks\n\nAn equilibrium point with both arrows toward it is stable, or a sink. With both arrows away from it, it is unstable, or a source. With one arrow toward it and one away, it is semi-stable.\nFor $f$ locally Lipschitz, a solution stays in one interval between equilibrium points and is monotone there.\nFor the [[logistic equation|0CC4VC7KR3]] $x' = x(1 - x)$, the point $0$ is a source and the point $1$ is a sink.\n\n## Explanations\n\n### L1 human\n\nThe phase line is a one-dimensional map of where the state can go.\nArrows show whether the state increases or decreases, and the equilibrium points are the places where it stops.\n",
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