A Discussion About Bursting Dynamics Induced By Pulse-shaped Explosion

Coupling effect of multiple time scale,as an important part of nonlinear dynamics,is ubiquitous in almost every field of engineering and science.Exploring the complex dynamical behaviors of multi-time-scale systems,especially the bursting oscillations,is one of the cutting-edge and hot issues in nowadays nonlinear field.In recent years,possible routes to bursting oscillations and the relevant dynamical mechanisms have been increasingly attracting attention of domestic and overseas scholars.Pulse-shaped explosion,reported recently,is a novel dynamical mechanism underlying the occurrence of bursting oscillations.In this thesis,taking the two-time-scale systems of Rayleigh’s type with multiple-frequency excitations as examples,we propose two new types of pulse-shaped explosion,i.e.,“positive and negative pulse-shaped explosion” and“bistable pulse-shaped explosion”,based on frequencies transformed fast-slow analysis and numerical simulations.The dynamical mechanisms of the bursting oscillations related to positive and negative pulse-shaped explosion and bistable pulse-shaped explosion are investigated by means of transformed phase portrait as well as fast-slow analysis,and different bursting patterns,i.e.,bursting of point-point type,bursting of cycle-cycle type and amplitude-modulated bursting,are obtained.Besides,the route to amplitude-modulated bursting by multiple-frequency slow parametric modulation is proposed.The main research contents are given as follows:(1)The dynamical behaviors and related mechanisms of a parametrically and externally excited Rayleigh system with 2:1 frequency ratio are considered.Firstly,the fast subsystem with single slow variable is obtained by frequencies transformed fast-slow analysis.The analysis of the fast subsystem shows that both the equilibrium point attractor and the limit cycle attractor can exhibit pulse-shaped sharp quantitative changes in relation to the variation of system parameters.In particular,the pulse-shaped explosion revealed here contains two different peaks in positive and negative directions.Thus,it can be named as “positive and negative pulse-shaped explosion”.With the variation of the slow parameter,the trajectory may undergo sharp transitions betweenthe rest and active states related to positive and negative pulse-shaped explosion,and therein lies the generation of bursting.Then,two different bursting patterns,i.e.,bursting of point-point type and cycle-cycle type,induced by positive and negative pulse-shaped explosion are obtained.(2)The frequency ratios of the paradigmatic externally and parametrically excited Rayleigh oscillator are extended to n:1,and the influences of the initial phase difference between two slow excitations on the pulse-shaped explosion are investigated.The result shows that an initial phase difference-?/ 2 with odd frequency ratios may lead to the coexisting of two equilibrium point attractors or limit cycle attractors.And the two coexisting solution branches can exhibit pulse-shaped explosion.That is to say,the bistable pulse-shaped explosion can be observed.The initial phase difference plays an important role in the pulse-shaped explosion and the transitions of the system between different attractors,which leads to the diversity of the dynamical behaviors of the fast subsystem,and finally results in complex bursting dynamics.Thus,two typical types of bursting patterns related to bistable pulse-shaped explosion are obtained.In addition,the pulse-shaped explosion can also induce the amplitude-modulated limit cycle attractor.Based on this,the route to amplitude-modulated bursting via pulse-shaped explosion is constructed.(3)The multiple-frequency slow parametric modulation(MFSPM)method,is proposed to obtain amplitude-modulated bursting.Similar to the pulse-shaped explosion of limit cycle sttractor,under certain conditions,the MFSPM leads to to-and-fro varyings of the amplitude of the traced active state(e.g.,limit cycle),which gives rise to oscillations in the envelope of the active phase,and finally creates amplitude-modulated bursting.The study shows that the evolution modes that the MFSPM exhibits do not depend on specific bifurcations.Thus,the MFSPM is the general method to obtain amplitude-modulated bursting.Then,the validity of the approach is demonstrated by several examples.Besides,the effects of the excitation frequencies on amplitude-modulated bursting are investigated.It is proved that the envelope of the amplitude-modulated bursting has the same oscillation frequency as that of theexcitation with relatively high frequency.