Research On Novel Bursting Patterns And Its Bifurcation Mechanisms

Things in nature are interrelated,influenced and restricted by each other.Different time scales may exist among different things.Analyzing the complex dynamic behavior and its mechanisms in multi-time scale coupled nonlinear system has become one of the cutting-edge and hot topics in the field of nonlinear science at home and abroad.It is widely used in biological science,chemical detection,machinery,medicine and other fields.Based on some important theories of nonlinear dynamics,such as the bifurcation theory,the fast-slow analysis method,the transformed phase portraits and the transformed fast-slow analysis method,we investigated the bursting dynamics in two time scale nonlinear dynamical system under the effect of multi-frequency slow excitations.The main works are given as following:1. We have studied the bursting dynamics of Sprott B system withmulti-frequency slow excitations,and obtained two novel bursting patterns,namely,the turnover-of-“Hopf/Hopf”-hysteresis induced bursting and cascaded“Hopf/Hopf”-hysteresis bursting.For the turnover-of-“Hopf/Hopf”-hysteresis induced bursting,distinct oscillations can be observed in the quasi-static process of bursting.Our study shows that the frequency of such large amplitude oscillations has the same frequency as the frequency of?₂.Interestingly,the number of“Hopf/Hopf”-hysteresis bursting each period increases as the increases of?2.Numerical simulations show that the number of“Hopf/Hopf”-hysteresis bursting each period equals to the ratio of?₂ to?₁.2. We have also studied the bursting dynamics of the smallest chemical reactionsystem with multi-frequency slow parametric excitations,and obtained four novel bursting patterns,namely,the turnover-of-transcritical-hysteresis induced bursting???,the cascaded transcritical-hysteresis bursting???,the turnover-of-transcritical-hysteresis/sup Hopf induced compound bursting???and cascaded“sup Hopf/sup Hopf”bursting through transcritical-hysteresis loop???.We investigated the generation mechanism of each bursting pattern and the transitions among different bursting patterns.We discussed the frequency nature of the large amplitude oscillations in the quasi-static process of the bursting patterns???and???,and explored the number of simple bursting pattern in cascaded bursting patterns???and???.3. Based on fast-slow analysis method,we investigated the dynamic mechanismsof above novel bursting patterns.We showed that,under the effect of the other excitation,both of the“Hopf/Hopf”-hysteresisis and transcritical-hysteresis exhibit two evolution modes,which have been found to be closely related to the generation of the novel bursting patterns.The first evolution mode,featuring many extreme points appearing on the hysteresis,results in the equilibrium branch becomes the one with twits and turns.Therefore,the turnover-of-hysteresis induced bursting patterns are obtained in the items 1 and 2.The other is related to the case when additional bifurcation points appear on the equilibrium hysteresis curves,which account for the generation of the cascaded hysteresis bursting in the items 1 and 2.