Hopf bifurcation

In bifurcation theory a Hopf or Andronov-Hopf bifurcation is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane. Under reasonably generic assumptions about the dynamical system, we can expect to see a small amplitude limit cycle branching from the fixed point. The limit cycle is orbitally stable if a certain quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical.

The normal form of a Hopf bifurcation is:

${\frac {dz}{dt}}=z(\lambda +b|z|^{2}),$

where $z,b$ are both complex and $\lambda$ is a parameter. Write

$b=\alpha +i\beta \,$.

The number $\alpha$ is called the first Lyapunov coefficient.

• If $\alpha$ is negative then there is a stable limit cycle for $\lambda >0$:

$z(t)=re^{i\omega t}\,$

where $r={\sqrt {-\lambda /\alpha }}$ and $\omega =\beta r^{2}$. The bifurcation is then called supercritical.

• If $\alpha$ is positive then there is an unstable limit cycle for $\lambda <0$. The bifurcation is called subcritical.

Hopf bifurcations occur in the Hodgkin-Huxley model for nerve membrane, the Belousov-Zhabotinsky reaction, the Lorenz attractor and in the following simpler chemical system called the Brusselator as the parameter $B$ changes:

${\frac {dX}{dt}}=A+X^{2}Y-(B+1)X$

${\frac {dY}{dt}}=BX-X^{2}Y$

The "Smallest Chemical Reaction System with Hopf Bifurcation" was found 1995 in Berlin, Germany.

References

• Steven H. Strogatz, "Nonlinear Dynamics and Chaos", Addison Wesley publishing company, 1994.
• Yuri A. Kuznetsov, "Elements of Applied Bifurcation Theory", Springer-Verlag, 2004, New York. ISBN 0-387-21906-4

Links

• Reaction-diffusion systems
• The Hopf Bifurcation
• Andronov-Hopf bifurcation page at Scholarpedia