Hopf bifurcation

A Hopf bifurcation or Hopf - Andronov bifurcation is a type of local bifurcation in nonlinear systems. It is named after the German - American mathematician Eberhard Frederich Ferdinand Hopf.

Hopf bifurcations are characterized in that when the variation of a parameter, an equilibrium will lose its stability and merges into a limit cycle. Depending on whether this limit cycle is stable or unstable, it is called a supercritical or subcritical Hopf bifurcation.

In a Hopf bifurcation crosses at an equilibrium point ( fixed point ) of the system is a pair of conjugate complex eigenvalues ​​resulting from the linearization of the system Jacobian matrix of the imaginary axis of the complex plane, the bifurcation point, the conjugated eigenvalues ​​are therefore purely imaginary.

The codimension of the Hopf bifurcation is the same as in the Saddle -node bifurcation, the pitchfork bifurcation and the Trans- critical bifurcation equal to one, the other types of bifurcations of codimension 1, are characterized by an eigenvalue of the Jacobian matrix at the fixed point.