A Discussion About Several Complex Bursting Structures In Discrete Systems

Coupling effects of multiple time scale are ubiquitous in both science and engineering,while investigation on the coupling mechanisms of different time scales and related complex dynamic behaviors,is one of the cutting-edge and hot topics in nowadays multiple time scale nonlinear dynamics.In recent years,multiple time scale dynamics based on discrete systems,especially the bursting phenomenon therein,has received more and more attentions by researchers in different fields.However,up till now,most of bursting patterns that have been revealed possess simple structures,which means that there is only one bifurcation in those bursts,while related spiking states are periodic.In this dissertation,aiming at several classical nonlinear discrete-time systems,analytical methods such as fast-slow analysis,frequencies transformed fast-slow analysis and so on,are used to explore the complex dynamics and bifurcation structures of several bursting patterns.What's more,with the help of bifurcation theories,classifications for different bursting patterns are conducted in detail,which gives a substantial expansion and supplement to existing classifications regimes.Based on that,several novel bursting behaviors such as“Fold/multiple inverse Flip”type,“cascade delayed Flip/cascade Flip”type,symmetric“chaos/period-2~n”type,symmetric“chaos/chaos”type and so on are obtained.The basic contents are given as follows:(1)Aiming at a class of one dimensional non-autonomous map,asymmetric bifurcation structures of fast sub-system are presented,thus the upper stable branch can evolve to chaotic attractor through Flip bifurcations,while the lower stable branch shows no Flip bifurcation.At the same time,the relative location of the Fold point and those Flip points has several possible schemes.Hence,when the slow variable reaches the critical value,there exist multiple fates of transition behaviors,which can be either periodic or chaotic,thus leads to different kinds of bursting patterns triggered by Fold bifurcation.(2)Based on the non-autonomous Duffing map,possible bursting behaviors when the trajectory just moves on single stable branch are discussed.The result shows that when the slow variable can't pass through those Fold points,although there is no transition between the upper and lower branch,when the slow variable passes those Flip points successively,system shows distinct bifurcation delay phenomenons,thus lead to gusty transitions to orbits with higher period.Thereout,a class of asymmetric bursting triggered by delayed Flip bifurcation are obtained.(3)On account of the non-autonomous Duffing map,symmetric bursting patterns evoked by a class of special global bifurcations named crisis are investigated.With the help of time-escaped algorithm,attributions of fast subsystem's basin of attraction are simulated,which show that there exist critical values of boundary crisis.So when the slow variable goes through these critical values,here comes the symmetric transition to those periodic or chaotic attractors coexisting with the bygone chaotic attractors.Thus symmetric bursting triggered by boundary crisis are presented.(4)For map systems under multiple slow excitations with commensurate frequencies,it's feasible to convert it into a equivalent system with single slow variable,thus the related frequencies transformed fast-slow analysis is derived.After that,application of this method on the chaotic Rulkov map with double excitations is conducted,which shows that variation of frequency ratio leads to different patterns of combined transition,which mean the diversity of corresponding bursting structures.Results here enrich the mechanism of multiple scale dynamics in map system,meanwhile it give a complement and expansion about orthodox fast-slow analysis.