Transcritical bifurcation

The transcritical bifurcation describes a process in which the stability ( " attractive " or " repulsive " ) of two equilibrium positions of a system is reversed.

It is thus a particular type of bifurcation of a nonlinear system.

The normal form of the transcritical bifurcation is:

Where the Bifurkationsparameter is.

The transcritical bifurcation has the following equilibrium points:

You can also take in the normal form of a (that is, you were disturbing the fixed point ) and neglecting the term, one obtains

For the temporal development of the disorder.

Thus for a stable fixed-point (i.e., the interference decreases over time from ) and unstable ( the error increases ). For it is exactly the opposite.

At the critical value of Bifurkationsparameters fixed point is the indifferent stable ( only one in this case).

Discrete system

For a discrete system of (I):

The location of the fixed points remains unchanged from the continuous system.