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 skies bifurcation in reference to the sudden creation of two fixed points.^[1]

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.^[2]

Saddle node bifurcation

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 topological 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.^[3]

Example in two dimensions

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 $\alpha$ ,

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),.

A saddle-node bifurcation also 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 $x$ .

Notes

↑ Strogatz 1994, p. 47.
↑ Kuznetsov 1998, pp. 80–81.
↑ Kuznetsov 1998, Theorems 3.1 and 3.2.

References

Kuznetsov, Yuri A. (1998), Elements of Applied Bifurcation Theory (Second ed.), Springer, ISBN 0-387-98382-1 .
Strogatz, Steven H. (1994), Nonlinear Dynamics and Chaos, Addison Wesley, ISBN 0-201-54344-3 ..
Weisstein, Eric W., "Fold Bifurcation", MathWorld.

Saddle-node bifurcation

Normal form

Example in two dimensions

See also

Notes

References