Universal classification of bifurcation solutions to a primary parametric resonance in van der Pol-Duffing-Mathieu's systems (Q1919001)

The equation \(\ddot x + x + (B + Cx^2 + Dx^4) \dot x + Ex^3 + Fx^5 + Gx \cos 2 \omega t = 0\), is analyzed, where \(B, C, D, E, F\) and \(G\) are parameters. Using tools in equivariant singularity theory the authors classify, via topological equivalence, the \((Z_2 -)\) codimension bifurcations greater than 3, according to the variation of the parameter values. The question of whether the approximate solutions from the classical perturbation can be topologically equivalent in describing the periodic responses and the original systems is discussed and a numerical simulation is exhibited at the end of the paper.