Study On Bursting Oscillations And The Dynamical Mechanism With Non-smooth Bifurcations

Non-smooth systems coupled with different scales have a wide range of practical engineering background.Non-smooth bifurcation can cause such coupled systems to exhibit complicated dynamic behaviors,which has become one of the hot topics in the field of nonlinear dynamics.In this dissertation,the dynamical behaviors and the generation mechanism of the bursting oscillationsof two scales coupled non-smooth systems have been investigated by using bifurcation theory of the nonlinear dynamics,Filippov's theory and numerical simulations,focusing on the effects of several classes of non-smooth bifurcationson bursting oscillations.The researches in this dissertation are mainly about two kinds of non-smooth systems.One is the system with piecewisesmooth and continuous vector field,i.e.,the piecewise-smooth continuous system and the other is the system with piecewise-smooth but discountinuous vector field,i.e.,the Filippov system.The basic contents of this disserta-tion are given as follows.(1)Based on the fourth-order Chua's circuit,a piecewise-smooth continuous system is established by introducing an external periodic excitation and a piecewise nonlinear resistor.A set of bursting oscillationsof the system are observed when there is a gap between the frequency of the external excitation and the natural one.Beside the conventional fold and Hopf bifurcations,non-smooth Hopf bifurcation and fold bifurcation of non-smooth limit cycle are also found existing in the fast subsystem.It is found that non-smooth Hopf bifurcation may lead to the generation of stable non-smooth limit cycle.The non-smooth Hopf bifurcation may lead to slow passage effect which is similar to the conventional Hopf bifurcation.Spiking states,induced by multiple spiking attractors which are separated by fold bifurcationsof the smooth and non-smooth limit cycles,may amalgamate into a larger spiking state.There may exist asymmetric bursting attractor in the symmetric non-smooth system.(2)Another piecewise-smooth continuous system is established by using a third-order Chua's circuit.By regarding the slow variable as a bifurcation parameter,bifurcationsof the fast subsystem are analyzed via theoritical and numerical methods.Three kinds of non-smooth bifurcationsare observed in the fast subsystem,which are nonsmooth Hopf bifurcation,C-bifurcation and fold bifurcation of non-smooth limit cycle.The C-bifurcation is caused by the collision of stable limit cycles caused by the non-smooth Hopf bifurcation and conventional supercritical Hopf bifurcation at the switching manifold.The existence of delayed C-bifurcation is also found in the study.The delayed C-bifurcation may lead to multiple transition patternsof trajectories,including two uncommon cases which are the transition patternsof jumping in reserve direction,i.e.from spiking state to the neighborhood of fold bifurcation point and to the neighbourhood of Hopf bifurcation point respectively.(3)Based on the third-order Chua's circuit,a Filippov system involving a switching manifold is established.Under the conditionsthat the boundary equilibrium bifurcation is of a persistence type,we get a non-smooth limit cycle bifurcation called boundary homoclinic bifurcation and a fold bifurcation of non-smooth limit cycle possessing a sliding segment.Research found that only if the slow-varying variable passes through the boundary homoclinic bifurcation point,can bursting oscillationsbe formed,and otherwise,the trajectories are absorbed by the pseudo-node of sliding vector field and approach to the pseudo-node.The quiescent state attractor of the system on the switching manifold can be made up of a single pseudo-equilibrium point,and the time of the system staying at the neighbourhood of this point can be obtained analytically.(4)We continue to study the Filippov system established in part(3)above.When the boundary equilibrium bifurcation of the Filippov system is of a non-smooth fold type,the loacl adding-sliding bifurcation,crossing-sliding bifurcation and fold bifurcation of non-smooth limit cycle which possesses a sliding segment and crosses the switching manifold are observed in the fast subsystem.Research found that the non-smooth fold bifurcation at the boundary equilibrium can lead to jumping behavior of the trajectory from the switching manifold to the attractors of the subsystem.The crossing-sliding bifurcation may not change the stability of the non-smooth limit cycle but change the topological structure of it,and therefore crossing-sliding bifurcation is not the significant bifurcation of transition between the spiking state and quiescent state.The local adding-sliding bifurcation of trajectory in the fast subsystem can cause adding-sliding bifurcation of limit cycle of the copuling system,and the adding-sliding structure will be contained in the bursting oscillationsbefore or after the occurrance of the adding-sliding bifurcations.