The work is devoted to the systematic research of the periodic, quasi-periodic or chaotic oscillations and the coexistence in the nonlinear Duffing-van der Pol type dynamical systems describing processes and phenomena in nature or engineering. The achieved result is the elaboration of the basic theory for searching nonlinear effects based on the concepts of periodic skeletons, bifurcation theory and complete bifurcation groups method. The main characteristics of each attractor are an amount and an order of bifurcation groups. The approach allows doing global analysis using the method of bifurcation groups for the driven Duffing-van der Pol oscillator under the change of the system parameters. It can also be used to forecast catastrophic situations and nonlinear paradoxical effects in the mechanics or other nonlinear dynamical systems with Duffing-van der Pol type oscillator.