Mixed mode oscillations induced by bi-stability and fractal basins in the FGP plate under slow parametric and resonant external excitations

In this paper, three patterns of mixed mode oscillations (MMOs) in the functionally graded piezoelectric (FGP) plate are analyzed. By introducing a slow axial excitation into the two degrees of freedom model, a fast-slow dynamic system with slow parametric and resonant external excitation is established. Using the resonance relations and the multiple scales method, the averaged equations as well as the approximation of stationary responses of the fast sub-system (FSS) are obtained. Thus, predictions of the possible distribution of periodic solutions are given for both two modes. Then numerical bifurcation diagrams of the FSS with respect to the control parameter are computed. It is shown that the FSS can present the coexistence of the stable limit cycle with small amplitude and the multi-period cycle with large amplitude. What is more, the amplitude of the external excitation has little qualitative effects on the bifurcation structure of the FSS. On the other hand, for different amplitude of the external excitation, three patterns of MMOs in the fast-slow system are observed. To explain the corresponding generation mechanisms, fast-slow analysis and basins of attraction (BOAs) of the FSS are combined to depict the transition behaviors of the fast-slow flow. Our results show that, as the amplitude of the external excitation increases, in the vicinity of the small amplitude cycle, the basin of attraction (BOA) of the big amplitude cycle will expand. Meanwhile, since the BOAs possess fractal structures, destinations of these transitions can show three patterns, namely cycle with small amplitude, multi-period cycle with large amplitude, hybrid of small/big cycle. Based on these, generation mechanisms of three MMOs patterns, which can be named as “cycle-chaos-cycle” MMOs, “cycle-chaos-big cycle” MMOs, “cycle-chaos-hybrid of small/big cycle” MMOs, are explained from a numerical view.