Saddle-node bifurcation

The saddle -node bifurcation ( saddle node bifurcation English ), pleats bifurcation (English fold bifurcation ), tangent bifurcation (English tangent bifrucation ), limit point or turning point is a specific type of a bifurcation of a nonlinear dynamic system.

The normal form of the saddle -node bifurcation is

Where the Bifurkationsparameter is.

This normal form has for fixed points:

This means that there is no fixed point for, for exactly one fixed point and two otherwise. The first fixed point is stable (node ​​), the second unstable (saddle). At the bifurcation point collide and saddle node. If we consider a system with higher-order

So these terms do not affect in a sufficiently small neighborhood of the saddle -node point, the behavior of the system. That is, the system is locally topologically equivalent to the origin to the normal form. Generally, the bifurcation is characterized in that an eigenvalue of the Jacobian matrix of the dynamic system at a critical value of Bifurkationsparameters zero.