Global Nonlinear Dynamics Based on the Method of Complete Bifurcation Groups and Rare Attractors

The paper is devoted to the global bifurcation analysis of the models of strongly nonlinear forced or autonomous dynamical systems with one or several-degree-of-freedom by direct numerical and/or analytical methods. 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 illustrated using four archetypal forced dynamical systems with one degree-of-freedom. They are Duffing model with positional force f(x) = x + x^3, Duffing double-well potential driven system, pendulum driven system and piecewise-linear (bilinear soft impact) driven dynamical system