Saddle-node bifurcation

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 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 r0 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.

Notes

References

References

1. (2015). "Computational techniques in mathematical modelling of biological switches".
2. (2018). "Einstein's field equations as a fold bifurcation". Journal of Geometry and Physics.
3. (2019-08-01). "Time-dependent saddle–node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions". Physica D: Nonlinear Phenomena.