Mechanism Analysis Of Mixed-Mode Oscillations With Multiple Frequency Excitations

The problem in nonlinear systems with multiple time scale is one of the leading subjects and hot topics in nonlinear science.It has wide application background,and involved in various fields of engineering and technology.Coupling effects of multiple time scale will lead to more complex dynamic behavior of the system,especially the bursting phenomenon therein.Therefore,it is far-reaching significant to explore the bursting oscillation and its corresponding mechanism of bifurcation in multiple time scale coupled systems.In this dissertation,aiming at two classes of two-time scales coupled dynamical system under different conditions,evolution of the bursting oscillations of the system has been investigated and the main contents are as follows:A novel three-dimensional chaotic system with a parametric excitation is investigated.By fixing the system parameters at suitable locations which implies the exciting frequency is far less than the natural frequency,the coupling of two time scales in the system has been established.The whole exciting term is taken as a bifurcation parameter of the fast subsystem,of which stability and bifurcation behavior with the variation of the parameters are explored.The dynamic mechanism of bursting oscillation of the system is revealed by slow-fast analysis method.Researches show that when the slow-varying exciting term passes through the supercritical pitchfork bifurcation point periodically,the bifurcation delay behavior can be observed in the system.With the exciting amplitude increasing,the delay effect of bifurcation becomes more and more obvious.When the bifurcation delay terminates at different attractor regions,the system trajectory will eventually switch to different attractors,resulting in different forms of bursting oscillations,such as point-point bursting,point-cycle bursting,etc.In order to investigate the bifurcation mechanism of bursting oscillations in a nonlinear system with two slowly varying excitations,a relative simple Duffing's oscillator with two external periodic excitations is introduced as an example.by employing Moivre's equation,the two periodic exciting terms can be expressed as a slow-varying parameter.Using the method of slow-fast analysis,the mechanism of different bursting have been presented.It is found that with the variation of the exciting amplitudes,the structure of the equilibrium curves and bifurcation behavior may change,which results in different forms of bursting oscillations.Combining with the transformed phase portraits,it is pointed out that the equilibrium branches and therelated bifurcations may not only influence the structures of attractors,but also change the mechanism of the transformation between quiescent states and spiking states,which leads to different forms of bursting oscillations,such as periodic 2-fold bursting,periodic 6-fold bursting,etc.When a nonlinear coupled system with two slow periodic excitations,the dynamical behavior of the system,due to the differences between the ratios of the excitation frequency,may have great diversities.In order to investigate the effect of different frequency ratios on the bursting oscillation,a external excitation is introduced on the basis of the mathematical model in chapter three.In consideration of whether the frequency ratios are commensurate or incommensurate,different methods are applied to merge two periodic excitation terms into one which can be regarded as the bifurcation parameter of the fast subsystem respectively.It can be found that the equilibrium branch of the fast system as well as the period of system change significantly with the difference between frequency ratios,for which different types of bursting oscillations occur.By taking the fast-slow analysis method,the mechanism of different bursting oscillations is revealed with the overlap of the transformed phase portrait and the bifurcation diagram of the equilibrium.