Bursting Oscillations As Wellas The Bifurcation Mechanism Of Piecewise Systems With Two Scales

Due to widespread existence of non-smooth systems in practical engineering,the complicated dynamic behaviors and the generation mechanism have become one of the hot topics at home and abroad.The non-smooth system not only shows common bifurcation behaviors like the smooth system,but also manifests some special bifurcation behaviors,such as grazing bifurcation,corner-collision bifurcation,sliding bifurcation,etc.In present,most of the researches on the non-smooth systems just contain single time scale.However,a large number of practical systems may involve the coupling effects between multiple scales.Therefore,it is necessary to make a deep study into dynamics of non-smooth systems with the multiple scales.Especially,the coupling of different time scales can lead to bursting oscillations as well as transition mechanism of bifurcation with different modal oscillation.The main purpose of this dissertation is to explore the bursting oscillations as well as the mechanism in non-smooth piece-wise system with different scales.Taking the simple but typical Duffing oscillator as an example,by introducing the piece-wise controller on the state variable and taking suitable parameter values,when the excitation frequency is far less than the natural frequency,the whole periodic exciting term can be considered as a slow-varying parameter.Under the coupling between two scales,a piece-wise systems with single slow-varying parameter is established.Due to the existence of non-smooth boundary,the trajectory is divided into two different regions.The equilibrium points as well as the bifurcation characteristics of the relevant fast subsystem under different regions are obtained.And two typical cases with different forms of bursting oscillations are discussed.It is pointed out that,different dynamic behaviors and the mechanism of non-smooth bifurcation are derived when the trajectory passes across the non-smooth boundary.By introducing transformed phase portrait and combining different regions of the equilibrium branches as well as the related bifurcation characteristics of the governing subsystem,we are found the phenomenon of bursting oscillation and sliding movement around the non-smooth boundary by the governing subsystem alternates between different regions.Which can cause the system lead to the special non-smooth bursting oscillation behaviors and further reveals the bifurcation mechanism.Based on the previous research,by introducing another periodic exciting term,i.e.periodic parameter excitation term.When the periodic external exciting frequency and the periodic parameter exciting frequency are far less than the natural frequency,a piece-wise systems with double slow-varying parameter under the coupling between two scales is established.We can choose the most slow time scale as the bifurcation parameter of fast subsystem,by introducing the auxiliary parameter and changing two periodic exciting term into a slow-varying as a same slow-varying parameter.The equilibrium points as well as the bifurcation of the relevant fast subsystem are analyzed.The influence of the frequency ratio of external exciting term and parameter exciting term on the system's dynamic behaviors is considered.As the frequency ratio increase,the bursting oscillation is more and more complex,which may behave not only adding the oscillating times obviously,but also producing more bifurcation modes,especially the form of motion at the non-smooth interface.By employing relate transformed phase portrait,it is pointed out that the distribution of the equilibrium branches of the governing subsystem influences the bursting oscillation.Which may lead to system through the non-smooth surface,may return to the original region instead of crossing the interface,and also reveals the bifurcation mechanism of the complex bursting oscillations.