Supercritical and subcritical Hopf bifurcationsEdit
Dynamics of the Hopf bifurcation near . Possible trajectories in red, stable structures in dark blue and unstable structures in dashed light blue. Supercritical Hopf bifurcation: 1a) stable fixed point 1b) unstable fixed point, stable limit cycle 1c) phase space dynamics. Subcritical Hopf bifurcation: 2a) stable fixed point, unstable limit cycle 2b) unstable fixed point 2c) phase space dynamics. determines the angular dynamics and therefore the direction of winding for the trajectories.
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.
Write: The number α is called the first Lyapunov coefficient.
If α is negative then there is a stable limit cycle for λ > 0:
The bifurcation is then called supercritical.
If α is positive then there is an unstable limit cycle for λ < 0. The bifurcation is called subcritical.
Normal form of the supercritical Hopf bifurcation in Cartesian coordinates.
The normal form of the supercritical Hopf bifurcation can be expressed intuitively in polar coordinates,
where is the instantaneous amplitude of the oscillation and is its instantaneous angular position. The angular velocity is fixed. When , the differential equation for has an unstable fixed point at and a stable fixed point at . The system thus describes a stable circular limit cycle with radius and angular velocity . When then is the only fixed point and it is stable. In that case, the system describes a spiral that converges to the origin.
The polar coordinates can be transformed into Cartesian coordinates by writing and . Differentiating and with respect to time yields the differential equations,
The normal form of the subcritical Hopf is obtained by negating the sign of ,
which reverses the stability of the fixed points in . For the limit cycle is now unstable and the origin is stable.
The Hopf bifurcation in the Selkov system (see article). As the parameters change, a limit cycle (in blue) appears out of a stable equilibrium.
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 bifurcationEdit
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 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point . Suppose that all eigenvalues of 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 ) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea.
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