Complete bifurcation analysis of pendulum vibration absorber and rare attractors

The methods and approaches of the global bifurcation analysis of nonlinear dynamical systems are under consideration. The main idea of new approaches is a concept of complete bifurcation groups, periodic skeletons, protuberances and rare attractors (RA). The method of complete bifurcation groups (MCBG) allows obtaining new qualitative results in well-known dynamical models. MCBG is based on the method of stable and unstable periodic regimes continuation on a parameter. New results obtained by the method are discussed in this paper for the pendulum vibration absorbers with one and several degrees-of-freedom. The concept of RA and protuberances illustrated by typical bifurcation groups allows finding new important applications for absorbers of vibration. Obtained results are important for explanations of some catastrophic or good-luck events in engineering.