# 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:

Here is the state variable and is the bifurcation parameter.

- If there are two equilibrium points, a stable equilibrium point at and an unstable one at .
- At (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 there are no equilibrium points.
^{[2]}

In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation which has a fixed point at for with is locally topologically equivalent to , provided it satisfies and . The first condition is the nondegeneracy condition and the second condition is the transversality condition.^{[3]}

## Example in two dimensions

An example of a saddle-node bifurcation in two dimensions occurs in the two-dimensional dynamical system:

As can be seen by the animation obtained by plotting phase portraits by varying the parameter ,

- When is negative, there are no equilibrium points.
- When , there is a saddle-node point.
- When 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 to , that is, the consumption rate is constant and not in proportion to resource .

Other examples are in modelling biological switches (see a tutorial for the computational techniques in modelling biological switches with an easy to understand synthetic toggle switch that demonstrates the bistability and hysteresis behavior showing the saddle-nodes or tipping points^{[4]}).

## See also

## Notes

- ↑ Strogatz 1994, p. 47.
- ↑ Kuznetsov 1998, pp. 80–81.
- ↑ Kuznetsov 1998, Theorems 3.1 and 3.2.
- ↑ Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf

## 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*.

- Chong, K. H.; Samarasinghe, S.; Kulasiri, D.; Zheng, J. (2015),
*Computational Techniques in Mathematical Modelling of Biological Switches*, In Weber, T., McPhee, M.J. and Anderssen, R.S. (eds) MODSIM2015, 21st International Congress on Modelling and Simulation (MODSIM 2015). Modelling and Simulation Society of Australia and New Zealand, December 2015, pp. 578-584, ISBN 978-0-9872143-5-5.