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.
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:
- <math>\frac{dx}{dt}=\mu+x^2</math>
Here <math>x</math> is the state variable and <math>\mu</math> is the bifurcation parameter.
- If <math>\mu<0</math> there are two fixed points, a stable fixed point at <math>-\sqrt{-\mu}</math> and an unstable one at <math>+\sqrt{-\mu}</math>.
- At <math>\mu=0</math> (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 <math>\mu>0</math> there are no fixed points.
A saddle-node bifurcation occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from <math>px</math> to <math>p</math>, that is the consumption rate is constant and not in proportion to resource <math>x</math>.
Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.
[edit] Example
An example of a saddle-node bifurcation in two-dimensions occurs in the two-dimensional dynamical system:
- <math> \frac {dx} {dt} = \alpha - x^2 </math>
- <math> \frac {dy} {dt} = - y.</math>
As can be seen by the animation obtained by plotting phase portraits by varying the parameter <math> \alpha </math>,
- When <math> \alpha </math> is negative, there are no equilibrium points.
- When <math> \alpha = 0</math>, there is a saddle-node point.
- When <math> \alpha </math> is positive, there are two equilibrium points: one attractor and one repellor, that is, one saddle and one node.
