ON THE BIRTH OF MULTIPLE LIMIT CYCLES IN NONLINEAR SYSTEMS

Abstract

Degenerate (or singular) Hopf bifurcations of a certain type determine the appearance of multiple limit cycles under system parameter perturbations. In the study of these degenerate Hopf bifurcations, computational formulas for the stability indexes (i.e., curvature coefficients) are essential. However, such formulas are very difficult to derive, and so are usually computed by different approximation methods. Inspired by the feedback control systems methodology and the harmonic balance approximation technique, higher-order approximate formulas for such curvature coefficients are derived in this paper in the frequency domain setting. The results obtained are then applied to a study of nonlinear dynamical systems within the region of one periodic solution, bypassing a direct investigation of the multiple limit cycles and some tedious discussion of the complex multiplicity issue. Finally, we will show that several types of stability bifurcations can be controlled based on the results obtained in this paper.