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a (local) bifurcation is a change in the number/type of equilibria in a parametrized dynamical system. in 1-D continuous-time, it's helpful to plot state (x) versus parameter (mu) to see the "multiverse" of potential dynamics as a function of parameter. 1/6
these local changes can be classified. most common is the "saddle-node" bifurcation, in which two equilibria of opposite stability coalesce and annihilate as you change the parameter. this one shows up all over the place. 2/6
next up: the "transcritical" bifurcation, in which two equilibria collide and exchange stabilities. it's not as common, since it requires a certain symmetry, but it's pretty useful. hey, did you notice the two transcriticals in the first video on this thread? 3/6
the last elementary type in 1-D itself splits into two sub-types. a "supercritical pitchfork" bifurcation is when a stable equilibrium goes unstable and sheds a symmetric pair of stable equilibria. this one is fairly common in physical systems with a symmetry. 4/6
...but it's dual, the "subcritical pitchfork", in which a symmetric pair of unstable equilibria collapse into a stable equilibrium, causing it to go unstable, is genuinely distinct, and genuinely dangerous! you might think you're stable, then poof! 5/6
there's so much more to bifurcation theory! the first big lesson to glean is that Taylor expansion is the key to identifying and classifying local changes in dynamical systems, as you'll see if you go back to the first video in the thread. 6/6
