{
  "aliases": [
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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f = (f_1, \\ldots, f_n) : U \\to \\mathbb{R}^n$ be a function, and consider the [[autonomous system|07BNRFQ0PJ]] $x_j' = f_j(x_1, \\ldots, x_n)$ for $j = 1, \\ldots, n$.\nFor $j \\in \\{1, \\ldots, n\\}$, the $j$-th nullcline of the system is the set\n$$N_j = \\{x \\in U : f_j(x) = 0\\}.$$\n\n## Remarks\n\nOn the $j$-th nullcline, the velocity has no component in the direction $x_j$. In the plane, trajectories cross the $x$-nullcline vertically and the $y$-nullcline horizontally.\nThe [[equilibrium points|0G4X36AE0X]] are the points that lie on all the nullclines.\nThe nullclines divide the [[phase plane|05VDGH129F]] into regions where each component of the velocity has a constant sign.\nFor a planar linear system, the nullclines are lines through the origin, or the whole plane when a row of the matrix is zero.\n\n## Explanations\n\n### L1 human\n\nA nullcline is the set of states where one of the variables stops changing for an instant.\nDrawing the nullclines is a quick first step to sketch a phase portrait.\n",
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