Using characteristic loci in the Hopf bifurcation

The Hopf bifurcation theorem is used by physicists, biologists and others interested in oscillations in systems described by differential equations. This paper presents a version of the theorem applicable directly to feedback systems, using the method of characteristic loci in an interpretation which looks rather like a multiple-loop version of the graphical describing function method. Frequency, amplitude and stability of oscillations about an equilibrium state are predicted by an iterative process which starts with a 2nd-order harmonic balance and successively increases the number of harmonics taken into account. The results from each stage of the process are used to define a curve in the complex plane from which the results of the next stage can be read off; consequently, the user has a visual indication of the accuracy and rate of convergence. The main theoretical limitation is that the nonlinear elements in the system must be continuously differentiable at least once more than the order of the highest harmonic used, whereas the main practical limitation is that the formula for the curve becomes extremely complicated if more than about six harmonics are considered.