New concepts of nonlinear dynamics: protuberances, unstable periodic infinitiums and rare attractors

A new approach for the global bifurcation analysis for strongly nonlinear dynamical systems, based on the ideas of Poincaré, Birkhoff and Andronov, is proposed. The main idea of the approach is a concept of complete bifurcation groups and periodic branch continuation along stable and unstable solutions, named by the author as a method of complete bifurcation groups (MCBG). The article is widely illustrated using archetypal dynamical systems with one-degree-offreedom. Among them are: Duffing model (symmetrical, asymmetrical) with one and two potential wells, piecewise-linear systems with one and several potential wells, impact and pendulum system