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 Poincare´, 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 + x3, Duffing double-well potential driven system, pendulum driven system and piecewise-linear (bilinear soft impact) driven dynamical system (Eq. 1–4). x+¨bx+˙x+x3=h1coswt(1)x+¨bx−˙x+x3=h1coswt(2)x+¨bx+˙a1sin(πx)=h1coswt(3)x+¨bx+˙f(x)=h1coswt,(4)f(x)=c1xifx≤d1,c2x−(c2−c1)d1ifx>d1 This paper is a continuation of the author’s previous one [53] with new results such as new bifurcation groups, rare attractors (RA) and protuberances. Some new results for dynamical systems with several degrees-of-freedom, based on the method of complete bifurcation groups may be found in [46–52].
Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion
