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 skies 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).
A typical example of a differential equation with a saddle-node bifurcation is:
Here is the state variable and is the bifurcation parameter.
- If there are two equilibrium points, a stable equilibrium point at and an unstable one at .
- At (the bifurcation point) there is exactly one equilibrium point. At this point the fixed point is no longer hyperbolic. In this case the fixed point is called a saddle-node fixed point.
- If there are no equilibrium points.
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 topological equivalent to , provided it satisfies and . 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 ,
- When is negative, there are no equilibrium points.
- When , there is a saddle-node point.
- When is positive, there are two equilibrium points: that is, one saddle point and one node (either an attractor or a repellor),.
A saddle-node bifurcation also occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from to , that is the consumption rate is constant and not in proportion to resource .