A Discussion About Three Novel And Complex Bursting Oscillation Patterns

Bursting oscillations exist spontaneously among various fields of science and engineering,and the complexity as well as the mechanisms of which have always been vital to bursting research.This paper,by considering some typical nonlinear systems,reveals complex bursting dynamics and their corresponding mechanism.With fast-slow analysis,frequency-transformation fast-slow analysis,bifurcation theory and numerical simulation methods being applied,several types of bursting oscillations,such as ‘turnover-of-pitchfork-hysteresis-induced bursting’,‘series-mode pitchfork-hysteresis bursting oscillations’ and ‘compound series-mode pitchfork-hysteresis bursting’etc.,can be obtained.And based on which novel ways to complex bursting dynamic patterns can be found.The main contents are as follows:(1)For a classic nonlinear Duffing system demonstrated with multiple-frequency parametric excitations,sample bursting behavior induced by pitchfork-hysteresis bifurcation have been studied in recent papers.Based on this we consider the Duffing system with multiple-frequency parametric excitations on Chapter 2,for which the complex bursting oscillations pattern may be triggered by complex excitation patterns.The result can be shown,by numerical simulation,that the large amplitude oscillation of bursting increases along with increasing the ratio of excitation frequency.Furthermore,we find that an increasing number of extreme points are created on the stable branches of the nonzero equilibria of pitchfork bifurcation structure by means of increasing ratio of excitation frequency,for which the turnover-of-pitchfork-hysteresis bifurcation occurs.In addition,the frequency characteristics of large amplitude oscillation in novel complex bursting is studied.(2)Frequency-transformation fast-slow analysis method plays an important role in studying the nonlinear systems with multiple-frequency slow varying excitation,whose advantage is obvious that it can reduce multiple slow variable of system to a single slow variable through De Moivre theorem.Based on the multiple-frequency slow variable excitation Duffing system same as that on the previous chapter,on Chapter 3,double cases that can be reduced simple and complex slow variable have been investigated.It is found that novel and complex bursting patterns occur in nonlinear system when increasing one of the excitation amplitude,namely,the large amplitude oscillation in periodic of bursting evolve from single burster to multiple bursters.Further bifurcation analysis shows that the number of bifurcation points in the fast subsystem grows when increasing of the excitation frequency.Each of bifurcation point,particularly,may induce a simple bursting pattern related to the pitchfork-hysteresis bifurcation,which is manifested in the presence of multiple bursters in each time series.We name this as ‘series-mode pitchfork-hysteresis burstingoscillation’.Further study shows that whether the bursting which induce by delayed pitchfork bifurcation can occur depends on the excitation amplitude.In the end of this section,we studied the relation between novel bursting patterns.(3)For those bursting patterns in recent research most of large amplitude oscillations are periodic patterns.Less of them,however,involve chaotic pattern.On chapter 4,we explore Lorenz system,which is the classic nonlinear chaotic system.With the influence of multiple-frequency slow varying excitation on the system taken into consideration,a complex bursting pattern with periodic bursters alternating with chaotic bursters,namely,‘Compound series-mode pitchfork-hysteresis bursting’ is studied.We obtained that there are three different regions in the fast subsystem with frequency-transformation fast-slow analysis which can be described as the periodic regions related to the periodic attractor,the chaotic regions related to the chaotic attractor and the coexistence region of the multiple attractors.When the slow variable goes though back and forth in these regions,the large amplitude oscillation in the bursters generates a novel pattern consisting of periodic bursters and chaotic bursters.Furthermore,we explore three patterns of different excitation frequency ratio.In the last section of this paper,we do the summary for those works above,and the future research directions are considered combined with the problems encountered in the study.