Hopf bifurcation

Complex eigenvalues ​​of any mapping (points). In the Hopf bifurcation, a pair of complex conjugate eigenvalues ​​crosses the imaginary axis.

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 or after Alexander Alexandrowitsch Andronow , who treated it with Witt and Chaikin in the Soviet Union in the 1930s. But the roots of the theory go back to Henri Poincaré at the end of the 19th century.

In a Hopf bifurcation crosses at an equilibrium point ( fixed point ) of the system a pair of complex conjugate eigenvalues selected from the linearization of the resulting system Jacobian the imaginary axis of the complex plane ; at the bifurcation point itself the conjugate eigenvalues ​​are purely imaginary . The Hopf bifurcations can only occur in two- or higher-dimensional systems, since the linearization of the system must have at least two eigenvalues ​​("a pair").

1: Supercritical Hopf bifurcation, 2: Subcritical Hopf bifurcation. Possible trajectories in red, stable structures in dark blue, unstable structures in dashed light blue.

The normal form of the Hopf bifurcation is

${\displaystyle {\frac {dz}{dt}}=z((\lambda +i)+(\alpha +i\beta )|z|^{2}).}$

It is

• ${\displaystyle z}$ a complex quantity
• t the time
• i the imaginary unit
• ${\displaystyle \lambda }$, and are real parameters ${\displaystyle \alpha }$ ${\displaystyle \beta }$
  • ${\displaystyle \lambda }$ is an intrinsic value.

Hopf bifurcations are characterized by the fact that, when a parameter is varied, an equilibrium loses its stability and changes to a limit cycle or a fixed point (see adjacent figure):

• In the case of the supercritical Hopf bifurcation ( ), a stable fixed point occurs that changes to an unstable fixed point or a stable limit cycle at the transition to . ${\displaystyle \alpha >0}$ ${\displaystyle \lambda <0}$ ${\displaystyle \lambda >0}$
• In the case of the subcritical Hopf bifurcation ( ), an unstable limit cycle or a stable fixed point occurs, which also changes into an unstable fixed point. ${\displaystyle \alpha <0}$ ${\displaystyle \lambda <0}$ ${\displaystyle \lambda >0}$

The parameters and essentially determine the stability of the system , whereas the rotation of the trajectories and thus also the winding direction influences. ${\displaystyle \lambda }$ ${\displaystyle \alpha }$ ${\displaystyle \lambda =0}$ ${\displaystyle \beta }$

The codimension of the Hopf bifurcation is the same as with the saddle-knot bifurcation , the pitchfork bifurcation and the transcritical bifurcation ; these other types of bifurcations of codimension 1 are, however, characterized at the fixed point by an eigenvalue of the Jacobian matrix . ${\displaystyle \lambda =0}$

literature

• John Guckenheimer, Philip Holmes: Nonlinear oscillations, dynamical systems and bifurcations of vector fields , Springer, ISBN 0-387-90819-6
• Yu.A. Kuznetsov: Elements of Applied Bifurcation Theory , Springer, 3rd edition 2004
• Jerrold E. Marsden , M. McCracken: Hopf Bifurcation and its Applications , Springer 1976

Web links

• Article . In: Scholarpedia . (English, including references) (Yuri Kuznetsov, Andronov-Hopf Bifurcation)

Individual evidence

1. ^ Hopf, branching off a periodic solution of a differential system, reports of the mathematical-physical class of the Saxons. Akad. Wiss. Leipzig, Volume 94, 1942, pp. 1-22