Wikipedia / Saddle-node bifurcation
Wikipedia / 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 pointss (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.
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:
is the bifurcation parameter.
.
(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.
there are no equilibrium points.
In fact, this is a normal form is locally topologically equivalent. 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:
,
is negative, there are no equilibrium points.
, there is a saddle-node point.
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.
See also
Notes
References