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 sky 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).

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

Normal form
A typical example of a differential equation with 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.



In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation $$ \tfrac{dx}{dt} = f(r,x) $$ which has a fixed point at $$ x = 0 $$ for $$ r = 0 $$ with $$ \tfrac{\partial f}{\partial x}(0,0) = 0 $$ is locally topologically equivalent to $$ \frac{dx}{dt} = r \pm x^2 $$, provided it satisfies $$ \tfrac{\partial^2\! f}{\partial x^2}(0,0) \ne 0 $$ and $$ \tfrac{\partial f}{\partial r}(0,0) \ne 0 $$. The first condition is the nondegeneracy condition and the second condition is the transversality condition.

Example in two dimensions
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).

Other examples are in modelling biological switches. Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.