Complex Bursting Dynamics in the Cubic-Quintic Duffing-Van Der Pol System with Two External Periodic Excitations
Xindong Ma, Qinshen Bi, Lifeng Wang
Abstract: This paper investigates the complex bursting behaviors of the cubic-quintic Duffing-van der Pol system with two slow-varying forcing periodic excitations. Four novel bursting patterns, namely turnover-of-fold-hysteresis-induced bursting, cascade-turnover-of-fold-hysteresis bursting, supHopf/supHopf bursting via turnover-of-fold-hysteresis loops and cascade supHopf/supHopf bursting via cascade-turnover-of-fold hysteresis loops, are studied based on the time domain waveform analysis, two-parameter bifurcation diagrams and overlay maps. We find that the fold/fold hysteresis loop plays an important role in the generation of the complex bursting characteristics when the second external periodic excitation is introduced. In particular, there are some ups and downs on the stable equilibrium branches, resulting in the emergence of the specific oscillations. These oscillations can be regarded as the spiking states in the turnover-of-fold- hysteresis-induced bursting and supHopf/supHopf bursting via turnover-of-fold-hysteresis loops. In addition, it is noticeable that the stable equilibrium branches can pass through the fold and supercritical Hopf bifurcation points many times in each bursting period, leading to the cascade-turnover-of-fold-hysteresis bursting and cascade supHopf/supHopf bursting via cascade-turnover-of-fold hysteresis loops. We also present a fact that the amplitude of the excitations play an important role in the dynamical evolutions among the four bursting oscillations. Finally, the correctness of our study is tested and verified by the numerical simulations.
文章链接：https://link.springer.com/article/10.1007/s11012-022-01544-2
