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 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point of it. Suppose that all eigenvalues of have negative real parts 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.
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