The Study Of Bifurcation Phenomena And Bursting Oscillation Of A Class Of Nonlinear Systems

Problems in engineering are closely related to nonlinear vibrations,and among them,multi timescale problems give rise to the complex dynamics phenomenon of bursting oscillations.Therefore,the study of bursting oscillations can provide guidance for practical engineering and is of great significance.With the refinement of engineering design,the effects of time delay are considered in many systems,so the study of time delay can better predict the dynamical behaviors of the system.In this paper,the bifurcation phenomenon and the behavior of bursting oscillations for three types of systems are studied.In the first part,this paper mainly introduced the background and significance of the research and the work of scholars in recent years.In the second part,the time delay Duffing system is analyzed by bifurcation theory,also,the related pitchfork bifurcation and fold bifurcation are obtained.It is found that the system generates pitchfork bursting oscillations and Fold/Fold bursting oscillations,where Fold/Fold bursting can be constructed by different paths.The phase difference between the parametric excitation and the external excitation has a crucial effect on the realization of the bursting oscillations.The effect of the system with and without time delay is compared,and Hopf bifurcation does not occur for positive time delay.In the third part,two-degree-of-freedom coupled van der Pol oscillators with time delay feedback are discussed by normal form and central manifold theories.It is found that the change of time delay can cause the switch of system stability,and there are rich dynamical behaviors near the double Hopf bifurcation point,including amplitude death region,limit cycle region and so on.In the fourth part,the bifurcation theory is used to analyze the multi-valued response phenomenon,the bursting oscillation phenomenon and its mechanism of the series multistable piezoelectric energy harvester system.On the one hand,it is found that the five stable system has a richer multi-valued response,and it exhibits different types of bursting oscillations by low frequency excitation,which is related to multiple single-valued response regions.On the other hand,the inverse piezoelectric effects due to bursting oscillations and the chaotic behaviors of the system subjected to high-low frequency excitation are analyzed.The above results are all verified numerically,and the results show that the theoretical analysis can reveal the mechanism behind the complex phenomenon to a certain extent.