Saddle-node bifurcation

In the mathematical area of bifurcation theory a saddle-node bifurcation or tangential 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:

::$frac\{dx\}\{dt\}=r+x^2$

Here $x$ is the state variable and $r$ is the bifurcation parameter.
*If there are two equilibrium points, a stable equilibrium point at $-sqrt\{-r\}$ and an unstable one at $+sqrt\{-r\}$.
*At $r=0$ (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 $r>0$ there are no equilibrium points.

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 $alpha$,

* When $alpha$ is negative, there are no equilibrium points.
* When $alpha\; =\; 0$, there is a saddle-node point.
* When $alpha$ is positive, there are two equilibrium points: that is, one saddle point and one node (either an attractor or a repellor),.

See also

*Pitchfork bifurcation

*Transcritical bifurcation

*Hopf bifurcation
*Saddle point