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periodic phenomena (vibrations, tremors, mood swings?) often arise from a [supercritical] Hopf bifurcation, in which a spiral sink becomes a spiral source, spawning an attracting limit cycle as the parameter passes the critical value
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seeing this in the full 3-d space (x-y-mu) reveals the limit cycles forming a paraboloid. this \sqrt{\mu} - growth in radius is why the bifurcation feels so sudden: the amplitude of oscillation is growing very quickly at first
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like pitchfork bifurcations, Hopf's come in supercritical & subcritical forms. in a subcritical Hopf, a repelling limit cycle fuses with a spiral sink to produce a spiral source. these are quite dangerous, since you think your world is stable, and then it spirals off...
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it's hard to overemphasize how common these are. if i get some free time (!) i'd like to start simulating some physical systems that have Hopf's -- fishtailing, helicopters, delayed bang-bang controls, neurons, + more... Hopf's are everywhere!
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