The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems

One of the most powerful methods for studying periodic solutions In autonomous nonlinear systems is the theory which has developed from a proof by Hopf. He showed that oscillations near an equilibrium point can be understood by looking at the eigenvalues of the linearized equations for perturbations from equilibrium, and at certain crucial derivatives of the equations. A good deal of work has been done recently on this theory and the present paper summarizes recent results, presents some new ones, and shows how they can be used to study almost sinusoidal oscillations in nonlinear circuits and systems. The new results are a proof of the basic part of the Hopf theorem using only elementary methods, and a graphical interpretation of the theorem for nonlinear multiple-loop feedback systems. The graphical criterion checks the Hopf conditions for the existence of stable or unstable periodic oscillations. Since it is reminiscent of the generalized Nyquist criterion for linear systems, our graphical procedure can be interpreted as the frequencydomain version of the Hopf bifurcation theorem.