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In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it 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 complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system,a small-amplitude limit cycle branches from the fixed point.
Contents
- Supercritical and subcritical Hopf bifurcations
- Remarks
- Example
- Definition of a Hopf bifurcation
- RouthHurwitz criterion
- Sturm series
- Propositions
- References
A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov.
Supercritical and subcritical Hopf bifurcations
The limit cycle is orbitally stable if a specific 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:
where z, b are both complex and λ is a parameter. Write
The number α is called the first Lyapunov coefficient.
Remarks
The "smallest chemical reaction with Hopf bifurcation" was found in 1995 in Berlin, Germany. The same biochemical system has been used in order to investigate how the existence of a Hopf bifurcation influences our ability to reverse-engineer dynamical systems.
Under some general hypothesis, in the neighborhood of a Hopf bifurcation, a stable steady point of the system gives birth to a small stable limit cycle. Remark that looking for Hopf bifurcation is not equivalent to looking for stable limit cycles. First, some Hopf bifurcations (e.g. subcritical ones) do not imply the existence of stable limit cycles; second, there may exist limit cycles not related to Hopf bifurcations.
Example
Hopf bifurcations occur in the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor and in the following simpler chemical system called the Brusselator as the parameter B changes:
The Selkov model is
The phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. See Strogatz, Steven H. (1994). "Nonlinear Dynamics and Chaos", page 205 for detailed derivation.
Definition of a Hopf bifurcation
The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a steady point is known as the Hopf bifurcation. The following theorem works with steady points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation phenomenon occurs.
Theorem (see section 11.2 of ). Let
Routh–Hurwitz criterion
Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea.
Sturm series
Let
The coefficients
Propositions
Proposition 1. If all the Hurwitz determinants
Proposition 2. If all Hurwitz determinants
The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition.
Example
Let us consider the classical Van der Pol oscillator written with ordinary differential equations:
The Jacobian matrix associated to this system follows:
The characteristic polynomial (in
The coefficients are:
The associated Sturm series is:
The Sturm polynomials can be written as (here
The above proposition 2 tells that one must have:
Because 1 > 0 and −1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if