Bursting Oscillations As Well As The Mechanism Analysis In A Four-dimensional Non-smooth System

Due to the far-ranging engineering application backgrounds of multi-timescale systems,with abundant nonlinear phenomena,especially bursting oscillations,their complex dynamic behaviors as well as mechanism analysis have been the hot spots in the field of nonlinear dynamics.In addition,non-smooth factors such as collision,switch,dry friction and the like,inevitably existing in actual engineering,have also received widespread concerns by scholars at home and abroad.In this dissertation,bursting oscillation behaviors with their generating mechanism of such multiple timescale coupling systems with non-smooth factors are investigated.In order to explore the bursting oscillations along with the bifurcation mechanism of a nonlinear dynamic system with single excitation,a new non-smooth system of two time-scale is obtained by introducing a non-smooth term and a periodic external excitation to a four-dimensional laser system.When there is a significant order gap between the natural frequency and the exciting one,the whole excitation term can be considered as a slow-varying parameter.Firstly,for two smooth subsystems,the stability of all equilibria as well as the critical conditions under those regular bifurcations,such as fold bifurcation(FB)and Hopf bifurcation(HB)are derived.Secondly,non-smooth bifurcations,for example,boundary equilibrium bifurcation,according to differential inclusion theory,are also explored via an auxiliary parameter.Several types of sliding bifurcations when the trajectory crossing the non-smooth interface,such as grazing-sliding bifurcation and multi-sliding bifurcation,could be observed with the slow variation.In addition,owing to the coexistence of stable attractors in the vector field,the symmetry breaking phenomenon of bursting attractors appears and the mechanism of which is finally revealed by overlapping the transformed phase portrait and bifurcation curves.Because of the complexity of time-scale effects and mechanism of a nonlinear system under multiple excitation couplings,a non-smooth system with parametric and external excitations is further explored based on the above researches.Firstly,by employing the Moivre formula,two excitation terms could be expressed as one slowlychanging parameter.The obvious fast-slow effects are represented when strictly adjusting both frequencies,making them much smaller than the natural one.Furthermore,the stability of equilibria and possible bifurcations in fast subsystem are easily judged with the help of fast and slow analysis method.Notably,the frequency ratio between two excitations is the focus of the study.It is found that fold and Hopf bifurcation points significantly increased and the structure of equilibrium diagrams become more complicated as the ratio increases,so that the trajectory also exhibits a variety of mixed-mode oscillations.The oscillations of the system also perform two symmetrical forms according to the two cases where the frequency ratio is integer or fractional.Finally,the effect of parameter variations on transition modes between spiking and quiescence states is also investigated.