Hopf bifurcation

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—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.
A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf.
Overview.
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:
Write: formula_2 The number "α" is called the first Lyapunov coefficient.
Example.
Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator and Classical electromagnetism. 
The Selkov model is
The phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. 
In railway vehicle systems, Hopf bifurcation analysis is notably important. Conventionally a railway vehicle's stable motion at low speeds crosses over to unstable at high speeds. One aim of the nonlinear analysis of these systems is to perform an analytical investigation of bifurcation, nonlinear lateral stability and hunting behavior of rail vehicles on a tangent track, which uses the Bogoliubov method.
Definition of a Hopf bifurcation.
The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed 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 formula_6 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point formula_7. Suppose that all eigenvalues of formula_6 have negative real part except one conjugate nonzero purely imaginary pair formula_9. A "Hopf bifurcation" arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.
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 formula_10 be Sturm series associated to a characteristic polynomial formula_11. They can be written in the form:
The coefficients formula_13 for formula_14 in formula_15 correspond to what is called Hurwitz determinants. Their definition is related to the associated Hurwitz matrix.
Propositions.
Proposition 1. If all the Hurwitz determinants formula_13 are positive, apart perhaps formula_17 then the associated Jacobian has no pure imaginary eigenvalues.
Proposition 2. If all Hurwitz determinants formula_13 (for all formula_14 in formula_20 are positive, formula_21 and formula_22 then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair.
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.
Consider the classical Van der Pol oscillator written with ordinary differential equations:
The Jacobian matrix associated to this system follows:
The characteristic polynomial (in formula_25) of the linearization at (0,0) is equal to:
The coefficients are:
formula_27 <br>
The associated Sturm series is:
The Sturm polynomials can be written as (here formula_29):
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 formula_32.