Universal classification of bifurcating solutions to a primary parametric resonance in van der Pol-Duffing-Mathicu's systems

摘要

The bifurcation of the second-order approximate solutions of nonlinear parametrically excited systems possessing generalized van der Pol's dampings and quintic Duffing's nonlinearities subjected to a primary parametric resonance is investigated. Using singularity theory with Z_2-symmetry, bifurcations of the solutions are universally classified in a topologically equivalent sense for Z_2-codimension≥3. The question of whether the approximate solutions from the classical perturbation methods can be topologically equivalent in describing the periodic responses and the bifurcations of the original systems is made dear. The numerical results indicate that the vibration characteristic may suddenly disappear in the range of Z_2-codimension≥4.