
Saddle-node bifurcation

In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) 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.
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.
Normal form.
A typical example of a differential equation with a saddle-node bifurcation is:
Here formula_2 is the state variable and formula_3 is the bifurcation parameter.
In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation formula_9 which has a fixed point at formula_10 for formula_11 with formula_12 is locally topologically equivalent to formula_13, provided it satisfies formula_14 and formula_15. The first condition is the nondegeneracy condition and the second condition is the transversality condition.
Example in two dimensions.
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 formula_18,
A saddle-node bifurcation also occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from formula_22 to formula_23, that is, the consumption rate is constant and not in proportion to resource formula_2.
Other examples are in modelling biological switches. Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.