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 r<0 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


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