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.

Saddle node bifurcation

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

Phase portrait showing saddle-node bifurcation

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 ,

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


  1. Strogatz 1994, p. 47.
  2. Kuznetsov 1998, pp. 80–81.
  3. Kuznetsov 1998, Theorems 3.1 and 3.2.
  4. Chong et al. (2015)


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