On the birth of multiple limit cycles in nonlinear systems

One type of degenerate (or, singular) Hopf bifurcations 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 obtain, and hence are usually obtained by different approximation techniques. In this paper, by using engineering feedback system methodology and harmonic balance approximation techniques, these formulas are derived in the frequency domain. The results obtained in this paper can be used to study a nonlinear system within regions of one periodic solution, instead of directly dealing with multiple limit cycles. Thus, the complex multiplicity of nonlinear system dynamical behavior can be avoided.