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 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).

The normal form of a saddle-node bifurcation is:

\frac{dx}{dt}=r%2Bx^2

Here x is the state variable and r is the bifurcation parameter.

A saddle-node bifurcation occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from px to p, that is the consumption rate is constant and not in proportion to resource x.

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

Example

An example of a saddle-node bifurcation in two-dimensions occurs in the two-dimensional dynamical system:

 \frac {dx} {dt} = \alpha - x^2
 \frac {dy} {dt} = - y.

As can be seen by the animation obtained by plotting phase portraits by varying the parameter  \alpha ,

See also

References

Weisstein, Eric W., "Fold Bifurcation" from MathWorld.