The main task of this article is to study the patterns of mixed-mode oscillations and non-smooth behaviors in a Filippov system with external excitation. Different types of periodic spiral crossing mixed-mode oscillation patterns, i.e., “cusp-F−/fold-F−” oscillation, “cusp-F−/two-fold/two-fold/fold-F−” oscillation and “two-fold/fold-F−” oscillation, are explored. Based on the analysis of the equilibrium and tangential singularities of the fast subsystem, spiral crossing oscillation around the tangential singularities is investigated. Meanwhile, by combining the fast and slow analysis methods, we can observe that the cusp, two-fold and fold-cusp singularities play an important role in generating all kinds of complex mixed-mode oscillations.

No 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.

By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.

This paper investigates the bursting oscillations of a externally and parametrically forced Rayleigh-Duffing oscillator, in which three intermittent bursting types and one normal bursting type, namely intermittent “supHopf/supHopf-supHopf/supHopf” bursting, intermittent “fold/Homoclinic-Homoclinic/supHopf” bursting, intermittent “fold/Homoclinic-supHopf/supHopf” bursting and “fold/Homoclinic” bursting, are analyzed respectively. Recognizing the excitations as slow-varying state variables, the corresponding autonomous system can be exhibited and the bifurcation characteristics is briefly investigated, in particular, the Homoclinic bifurcation is analyzed by means of the Melnikov criterion. This paper shows that the dynamical behaviors of the excited Rayleigh-Duffing oscillator is touchy to the chosen of system parameters, different parameter conditions lead to distinct bifurcation structures that result in the trajectory approaching to different stable attractors and the appearance of different bursting forms. Our study increases the variousness of bursting oscillations and deepens the cognition of the generation mechanism of bursting dynamics. Lastly, the accuracy of the analysis presented in this paper is fully vindicated by the numerical simulations.