Complex bifurcation structures and bursting oscillations of an extended Duffing‐van der Pol oscillator

Abstract

This work focuses on the compound bursting dynamics and their generation mechanisms in an extended Duffing‐van der Pol oscillator excited by parametrical and external slow‐varying excitations. By considering the cosine excitation as a slow‐changing variables and using the Melnikov criterion, the fold, Hopf, and Homoclinic bifurcation critical conditions of the generalized autonomous system are obtained, and the bifurcation sets separate the whole parameter plane into seven different areas. Based on that, five compound bursting patterns, i.e., compound “supHopf/supHopf” type, compound “fold/fold” type, compound “fold/Homoclinic” type via a “fold/fold” and two “fold/Homoclinic” hysteresis loops, compound “fold/supHopf‐supHopf/Homoclinic” type via a “fold/fold” and two “fold/Homoclinic” hysteresis loops and compound “fold/supHopf” type via three “fold/fold” hysteresis loops, are studied. In addition, the mechanism of a period motion is also investigated by overlapping the projection on the space of to the bifurcation diagrams. Our research shows that the bursting oscillations are sensitive to the parameter , especially for the small values of . Moreover, this work explicates that how the choice of the system parameters affects the trends of the slow manifolds. Finally, the numerical simulations are used to illustrate and test the correctness of this work.