Symmetric And Asymmetric Bursting Oscillations As Well As The Bifurcation Mechanism

Many fast-slow dynamical phenomena have been found in various regions and disciplines,i.e.,the oscillatory reaction in the chemistry,the periodic excitability of the cell membrane in the neuroscience and complex oscillations in the optoelectronics field.In the dissertation,our works focus on the fast-slow behavior in the nonlinear dynamical system having three vector fields.The innovative results will be given as follows:1?To investigate the motion of the vector fields driven by the external periodic excitation with a very small frequency,the vector fields is divided into two parts,the fast-subsystem and the slow-subsystem.Obviously,the slow-subsystem provides the forcing without any feedback from the fast-subsystem.It is a different situation distinguishing from the another situation that the fast-subsystem and the slow-subsystem interact with each other.Therefore,under the drive of the external excitation,the fast-subsystem will occur more novel and abundant patterns of the periodic bursters.2?When add the external excitation to the vector fields,the system will be the non-autonomous form because the excitation depend explicitly on time,which implies the forced system have no any equilibrium points with the change of the time.Fixing the time,there is some“instantaneous”equilibrium points.However,it results in error outcome because the sequence of these instantaneous”equilibrium points is not solution of the the non-autonomous vector fields.For the slow-varying property of the external excitation,it is feasible that the whole excitation is viewed as the controlling parameter w.Furthermore,the non-autonomous vector fields will be converted to the generalized autonomous system.Therefore,analysis on the equilibrium branches of fast-subsystem will be reasonable.3?The equilibrium branches of fast-subsystem have been changed in term of the quantity and location due to the influence of the controlling parameter w.Consequently,the procedure of the analysis with respect to the equilibrium points will be nontrivial.4?In the dissertation,using some effective techniques i.e.,the parameter extension approach,the center manifold theorem and the normal forms theory,we test and verify respectively the existing conditions of some bifurcations occurring in the vector fields forced by the external excitation.For instance,we analyze,in detail,the three conditions with respect to Andronov-Hopf bifurcation?HB?,and confirm that it is super-critical.Specially,by introducing a small perturbed parameter,we study the dynamical structure of Pitch-fork bifurcation?PB?nearing the neighborhood of the bifurcation point.Finally,by applying the center manifold theorem and the normal forms theory,we get the second order and third order critical normal forms of the Double Zero bifurcation?BT?.5?In the analysis on theZ₂ symmetric vector fields,we pay attention to the asymmetric bursters and the symmetric bursters,as well as the relationship between them.With the change of the amplitude of the external excitation,numerical simulation indicates that the asymmetric bursters will pass through the separatrix and interact with each other,which causes the formation of the symmetric bursters.According that property,we define some novel patterns of the bursting oscillations.In addition,for an type of asymmetric vector fields driven by the external excitation,it witnesses several types of the bursting oscillations passing the Double Zero bifurcation point.