Hybrid Methods for Studying Stability and Bifurcations in Delayed Feedback Systems

The dynamics of two related models of second order delay differential equations with four bifurcating parameters are analyzed. Through a classical technique in the time domain which involves the location of the roots of an exponential polynomial equation, the areas of stability of the equilibrium are set. A frequency-domain methodology is applied to study the Hopf bifurcation phenomena and to describe the behavior of the emerging cycles completely via a feedback system approach. Certain types of singularities, which provoke fold bifurcations of cycles are detected precisely. Also, a complete picture of parameter configurations to produce resonances is established for both models. All results are checked with the software DDE-Biftool.