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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 $\\mu$ near $\\mu_0$ the system has a chaotic [[attractor|7FWX24X5MW]] $A_\\mu$.\nA *crisis* at $\\mu_0$ is a bifurcation at which $A_\\mu$ changes discontinuously as $\\mu$ crosses $\\mu_0$.\nThe attractor is suddenly destroyed or suddenly jumps in size.\nAt a crisis, $A_{\\mu_0}$ touches an unstable [[periodic orbit|0AA97XA70K]] or its stable manifold.\nThe crisis is a *boundary crisis* if this orbit lies on the boundary of the basin of $A_\\mu$.\nIt is an *interior crisis* if this orbit lies inside the basin.\n\n## Remarks\n\nExample: the [[logistic map|QN9X9BNKTX]] $f_r(x) = r x (1 - x)$ at $r = 4$.\nFor $r = 4$, the interval $[0, 1]$ is invariant and carries a chaotic attractor.\nThe attractor contains the repelling fixed point $0$ and its preimage $1$, which lie on the boundary of the basin.\nFor $r \u003e 4$, points near $1/2$ are mapped above $1$ and then escape to $-\\infty$, so the attractor is destroyed: a boundary crisis.\nAfter the crisis, typical orbits stay near the old attractor for a long time before escaping, a chaotic transient.\nNon-example: a [[period-doubling bifurcation|NQDTTZ59X8]] of a periodic attractor is not a crisis, since the attractor changes continuously and is not chaotic.\nCrises are a basic route by which [[strange attractors|9WKCZGG906]] appear or vanish in [[bifurcation theory|D9TW46P19Y]].\n",
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