Global Bifurcation Analysis of a Duffing-Van der Pol Oscillator with Parametric Excitation

In this paper, a composite cell state space is constructed by a multistage division of the continuous phase space. Based on point mapping method, global properties of dynamical systems can be analyzed more accurately and efficiently, and any small regions can be refined for clearly depicting some special basin boundaries in this cell state space. Global bifurcation of a Duffing-Van der Pol oscillator subjected to harmonic parametrical excitation is investigated. Attractors, basins of attraction, basin boundaries, saddles and invariant manifolds have been obtained. As the amplitude of excitation increases, it can be observed that the boundary crisis occurs twice. Then a basin boundary with Wada property appears in the state space and undergoes metamorphosis in the chaotic boundary crisis. At last, two attractors merge into a chaotic one when they simultaneously collide with the chaotic saddle embedded in the fractal boundary. All these results show the effectiveness of the proposed method in global analysis.