Bursting Oscillations And Bifurcation Mechanisms Of Multi-time Scale Chaotic Systems

As an indispensable part of nonlinear dynamics,multi-time scale coupling has great value and development prospects in practical engineering applications.It is the current research focus of nonlinear dynamics.Multi-time scale factors will induce the system to produce abundant dynamic behaviors,among which cluster oscillation phenomenon is a typical representative.Therefore,it is of far-reaching significance to explore the cluster oscillation and its evolution mechanism in a multi-time-scale coupled system.This paper takes two types of fast and slow coupled systems as examples to analyze the evolution behavior of system dynamics under different conditions.The main contents are as follows:1.Construct a two-time-scale QI system driven by parameter excitation,use the fast-slow analysis theory,treat the parameter excitation as the slow-varying bifurcation parameter of the fast subsystem,and observe and analyze the stability and bifurcation of the fast subsystem's equilibrium point branch.The mode can clearly explain the underlying mechanism of system cluster oscillation.Studies have shown that when the slowly varying excitation periodically passes through the fork-shaped bifurcation point,the system exhibits a significant delay phenomenon,and as the excitation amplitude increases,the delay effect becomes more obvious.When the bifurcation delay behavior ends in different attractor regions,the trajectory of the system tends to be different attractor motions,which results in different types of cluster oscillations.2.Next,a parameter-driven Shimizu-Morioka system is proposed.When the slowly varying parameters periodically pass through the transcritical bifurcation point,an obvious time lag behavior is observed.The fast and slow analysis method is used to study the periodic and chaotic burst oscillations caused by this delay behavior.More interestingly,the fast system has a zero equilibrium point branch and two asymmetrically distributed equilibrium point branches about slowly varying parameters.Therefore,when the delay transcritical bifurcation occurs,there are two possible paths for the trajectory to choose from.This leads to different excitation cycles corresponding to the cluster hair mode,resulting in two hybrid cluster hair modes,namely asymmetric composite delay transcritical/transcritical-delayed transcritical/sup Hopf/fold cluster oscillation,asymmetric Compound type delayed transcritical/sub Hopf/sub Hopf/fold-delayed transcritical/sup Hopf/fold cluster oscillation.In addition,the influence of the excitation frequency on the delay interval is also considered.The results of the study found that when the excitation amplitude is fixed and the excitation frequency is changed,the delay behavior will end in different parameter regions,resulting in different burst modes.Finally,the validity of the research in this paper is verified by numerical simulation.