Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises.
[1] 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 as a parameter crosses a threshold value.
Under reasonably generic assumptions about the dynamical system, the fixed point becomes a small-amplitude limit cycle as the parameter changes.
The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical.
The normal form of a Hopf bifurcation is the following time-dependent differential equation: Write:
The number α is called the first Lyapunov coefficient.
The normal form of the supercritical Hopf bifurcation can be expressed intuitively in polar coordinates, where
The system thus describes a stable circular limit cycle with radius
In that case, the system describes a spiral that converges to the origin.
with respect to time yields the differential equations, and The normal form of the subcritical Hopf is obtained by negating the sign of
the limit cycle is now unstable and the origin is stable.
Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane potential,[4] the Selkov model of glycolysis,[5] the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator, and in classical electromagnetism.
[6] Hopf bifurcations have also been shown to occur in fission waves.
[7] The Selkov model is The figure shows a phase portrait illustrating the Hopf bifurcation in the Selkov model.
[8] In railway vehicle systems, Hopf bifurcation analysis is notably important.
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.
order, we would obtain three ordinary differential equations in
increases from negative to positive, the origin turns from a stable spiral point to an unstable spiral point.
Thus we find that the Hopf bifurcation creates an attracting (rather than repelling) limit cycle.
This is plotted in the illustration on the right.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.
be the Jacobian of a continuous parametric dynamical system evaluated at a steady point
have negative real part except one conjugate nonzero purely imaginary pair
A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.
Routh–Hurwitz criterion (section I.13 of [12]) gives necessary conditions so that a Hopf bifurcation occurs.
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.
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 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
