Saddle-node bifurcation

The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems.

Another name is blue sky bifurcation in reference to the sudden creation of two fixed points.

[1] If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node).

Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.

A typical example of a differential equation with a saddle-node bifurcation is: Here

In fact, this is a normal form of a saddle-node bifurcation.

A scalar differential equation

is locally topologically equivalent to

[3] An example of a saddle-node bifurcation in two dimensions occurs in the two-dimensional dynamical system: As can be seen by the animation obtained by plotting phase portraits by varying the parameter

, Other examples are in modelling biological switches.

[4] Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation.

[5] A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.