In the mathematical area of bifurcation theoryasaddle-node bifurcation, tangential bifurcationorfold bifurcation is a local bifurcation in which two fixed points (orequilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. 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 is the state variable and
is the bifurcation parameter.
In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation which has a fixed point at
for
with
is locally topologically equivalentto
, provided it satisfies
and
. The first condition is the nondegeneracy condition and the second condition is the transversality condition.[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.[6]