Saddle-node bifurcation

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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 fixed points is unstable (the saddle), while the other is stable (the node).

The normal form of a saddle-node bifurcation is:

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

Here $x$ is the state variable and $μ$ is the bifurcation parameter.

If $μ < 0$ there are two fixed points, a stable fixed point at $-\sqrt{-\mu}$ and an unstable one at $+\sqrt{-\mu}$ .
At $μ = 0$ (the bifurcation point) there is exactly one fixed 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 $μ > 0$ there are no fixed points.

A saddle-node bifurcation occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from $p x$ 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.

[edit] Example

Phase portrait showing Saddle-node bifurcation.

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 $α$ ,

When $α$ is negative, there are no equilibrium points.
When $α = 0$ , there is a saddle-node point.
When $α$ is positive, there are two equilibrium points: one attractor and one repellor, that is, one saddle and one node.