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.^[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:

${\displaystyle {\frac {dx}{dt}}=r+x^{2}.}$

Here ${\displaystyle x}$ is the state variable and ${\displaystyle r}$ is the bifurcation parameter.

• If ${\displaystyle r<0}$ there are two equilibrium points, a stable equilibrium point at ${\displaystyle -{\sqrt {-r}}}$ and an unstable one at ${\displaystyle +{\sqrt {-r}}}$.
• At ${\displaystyle r=0}$ (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 ${\displaystyle r>0}$ 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 ${\displaystyle {\tfrac {dx}{dt}}=f(r,x)}$ which has a fixed point at ${\displaystyle x=0}$ for ${\displaystyle r=0}$ with ${\displaystyle {\tfrac {\partial f}{\partial x}}(0,0)=0}$ is locally topologically equivalent to ${\displaystyle {\frac {dx}{dt}}=r\pm x^{2}}$, provided it satisfies ${\displaystyle {\tfrac {\partial ^{2}\!f}{\partial x^{2}}}(0,0)\neq 0}$ and ${\displaystyle {\tfrac {\partial f}{\partial r}}(0,0)\neq 0}$. 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:

${\displaystyle {\frac {dx}{dt}}=\alpha -x^{2}}$

${\displaystyle {\frac {dy}{dt}}=-y.}$

As can be seen by the animation obtained by plotting phase portraits by varying the parameter ${\displaystyle \alpha }$,

• When ${\displaystyle \alpha }$ is negative, there are no equilibrium points.
• When ${\displaystyle \alpha =0}$, there is a saddle-node point.
• When ${\displaystyle \alpha }$ 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.^[4] Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation.^[5] A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.^[6]

See also

• Pitchfork bifurcation
• Transcritical bifurcation
• Hopf bifurcation
• Saddle point