Bursting Oscillations And Mechanism Analysis Of Nonlinear Oscillators Under Slow Excitation

As an important part of nonlinearity,the multi-time scale coupled nonlinear dynamics expounds the complex mechanism of nonlinear dynamics from the direction of dynamics and has become one of the most popular research contents of nonlinear dynamics.This dissertation analyzes the dynamic phenomena of two time-scale couplings under three types of periodic excitations by using the frequency conversion fast-slow analysis method as the basic theory combined with bifurcation analysis as well as transformed phase portraits.The main related work and the specific contents are as follows:First,an asymmetric R?ssler chaotic system used as a prototype is transformed into a four-dimensional nonlinear fast-slow coupling symmetrical system with multi-time scales.The system becomes a generalized autonomous system by treating the periodic external excitation of the improved system as a slow variable.The corresponding bifurcation analysis is performed by using the fast-slow analysis method,and then the numerical simulation is performed to calculate the corresponding bifurcation conditions(bifurcation value).Then,three typical dynamic behaviors are given by changing the magnitude of the excitation amplitude under the condition of constant value parameters,several typical bursting oscillations obtained combined with tools such as time history,phase diagram,and transformed phase portraits to describe and reveal the periodic bursting oscillation behavior as well as the generation mechanism mode of the system.Secondly,for a four-dimensional coupling Matheu-Van der Pol chaotic system,the fast and slow dynamic behavior of the system is studied by introducing the parameter dynamics.The system exhibits multi-scale effects in different frequency domains due to the magnitude difference between the excitation frequency and the system's own frequency.When the corresponding system parameters are determined,the parameter sets of slow-varying parameters and bifurcation parameters can be obtained.Therefore,a bifurcation diagram as well as four typical bursting oscillation modes were determined.Specially,slow channel effect,a special phenomenon of the system trajectory,occurred due to the magnitude difference between the two parameters in the slow-fast dynamic system,which will cause the trajectory to jump to the stable limit cycle after a delay instead of generating a branch immediately when passing through the bifurcation point.Finally,discussing a coupling Van der Pol-Duffing oscillator circuit which becomes a generalized autonomous system by introducing the periodic external excitation and conducting relevant research,the bifurcation conditions are analyzed and the coupling effect between different scales in frequency domain is considered.The external excitation is regarded as a slowly changing parameter because the frequency of external excitation is far less than the natural frequency.Under certain parameters Hopf bifurcation,fold bifurcation,Hopf-Hopf bifurcation and fold bifurcation of limit cycles will be observed by analyzing the bifurcation behavior in different modes.It can be found that there are five different typical bursting oscillations due to the different conditions of bifurcation.The dynamic behavior is revealed by introducing time history,phase diagram,and transformed phase portraits and different bifurcation behaviors will cause the system to excite between spiking state and the quiescent state.