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  "body": "\n## Definition\n\nLet $(\\phi_\\mu)$ be a one-parameter family of dynamical systems, flows or maps, with $\\mu \\in \\R$.\nAssume that for $\\mu$ on one side of $\\mu_0$ the system has a stable [[periodic orbit|0AA97XA70K]], and that it disappears or loses stability at $\\mu_0$.\nThe system shows *intermittency* for $\\mu$ just past $\\mu_0$ if typical orbits alternate between two kinds of phases.\n\n1. Laminar phases: long stretches of time during which the orbit stays close to the former periodic orbit.\n2. Bursts: short irregular stretches, after which the orbit is reinjected near the former periodic orbit.\n\nThe *type* is I when the bifurcation at $\\mu_0$ is a [[saddle-node bifurcation|MAP5J6HK0T]].\nIt is II for a subcritical [[Hopf|F5H8GW1XYS]] or Neimark–Sacker bifurcation, and III for a subcritical [[period-doubling bifurcation|NQDTTZ59X8]].\n\n## Remarks\n\nExample: the [[logistic map|QN9X9BNKTX]] $x \\mapsto r x (1 - x)$ near $r_c = 1 + 2\\sqrt{2}$.\nAt $r_c$ a stable and an unstable orbit of period $3$ are born in a saddle-node bifurcation of the third iterate.\nFor $r$ slightly below $r_c$, orbits pass slowly through the narrow gaps left between the graph of $f^3$ and the diagonal, so they look nearly periodic of period $3$.\nBetween these passages they move chaotically: this is type I intermittency.\nFor type I the mean length of the laminar phases grows like $(r_c - r)^{-1/2}$.\nNon-example: just after a supercritical period-doubling bifurcation the new stable orbit of double period attracts nearby orbits, so there are no bursts.\nIntermittency is a route to chaos studied in [[bifurcation theory|D9TW46P19Y]].\n",
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