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  "body": "\n## Summary\n\nBifurcation theory studies how the qualitative behaviour of a family of [[dynamical systems|BBZP1R872N]] changes as parameters vary.\nA bifurcation happens at a parameter value where the [[phase portrait|0T3GK0XCBC]] changes its topological type, for example when [[equilibria|0N8NGKDZDZ]] appear, disappear or change stability.\nThe theory classifies the typical bifurcations, such as the [[saddle-node|MAP5J6HK0T]] and [[Hopf|F5H8GW1XYS]] bifurcations, and gives normal forms for them.\nIt explains sudden transitions in models from physics, biology and engineering.\n\n## Explanations\n\n### L1 human\n\nMany systems change smoothly when you turn a knob, but at certain settings their behaviour changes in kind.\nA pendulum with friction comes to rest; push it with a motor and past some strength it starts to swing forever.\nBifurcation theory finds these critical settings and describes what happens there.\nA simple example is $\\dot x = \\mu - x^2$: for $\\mu \u003c 0$ there is no resting state, and for $\\mu \u003e 0$ there are two, which are born together at $\\mu = 0$.\n",
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