Hopf bifurcation

In bifurcation theory a Hopf or Andronov-Hopf bifurcation is a local bifurcation in which a fixed point ofa dynamical system loses stability as a pair of complex conjugate
eigenvalues of the linearization around the fixed point crossthe imaginary axis of the complex plane. Under reasonably genericassumptions about the dynamical system, we can expect to see a smallamplitude limit cycle branching from the fixed point. The limit cycle is orbitally stable if acertain 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 aparameter. Write:: $b= alpha + i eta ,$ . The number $alpha$ is called the first Lyapunov coefficient.

* If $alpha$ is negative then there is a stable limit cycle for $lambda > 0$ ::: $z(t) = r e^{i omega t} ,$ :where $r=sqrt{-lambda/alpha}$ and $omega=eta r^2$ . 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 nervemembrane, the Belousov-Zhabotinsky reaction, the Lorenz attractor and in the following simpler chemical system called theBrusselator as the parameter $B$ changes::: $frac{dX}{dt} = A + X^2Y -(B+1)X$ :: $frac{dY}{dt} = B X -X^2Y$

Notes

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
* [http://www.egwald.com/nonlineardynamics/bifurcations.php#hopfbifurcation The Hopf Bifurcation]
* [http://www.scholarpedia.org/article/Andronov-Hopf_bifurcation Andronov-Hopf bifurcation page] at Scholarpedia

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