Analysis of static and dynamic bifurcations from a feedback systems perspective

In this paper, several static and dynamic bifurcation conditions for nonlinear dynamical systems are obtained from a feedback systems approach. The static bifurcations are related to the multiplicity of the steady-state solutions, while the dynamic bifurcations are connected to the appearance of periodic solutions from the equilibrium curve under the Hopf bifurcation mechanism. Examples containing elementary bifurcations are presented with an input-output description of the system, also known as the frequency-domain approach, involving applications of the generalized Nyquist stability criterion (GNSC) and harmonic balance techniques. Simple degenerate Hopf bifurcations - in a hand calculation example as well as in a more elaborated numerical problem-are also shown. The approach has a feedback characteristic flavour as compared to the classical results obtained in the traditional time-domain setting. Analysing Hopf bifurcations from such a feedback systems perspective is deemed to be a preliminary step to gain insights into more advanced research topics in controlling bifurcations.