Bifurcation Control Of A Class Of Non-smooth Dynamical Systems

In this paper,we study the control of near-grazing dynamics of a class of vibro-impact system with symmetrical constrains,and the bifurcation control of a class of pyramidal neurons with non-smooth characteristics for the dorsal cochlear nucleus(DCN).Both of these systems belong to the first class of non-smooth systems,namely,impulsive systems.The specific research content are as follows:Firstly,for the two-degree-of-freedom vibro-impact system with symmetric constraints,the near-grazing dynamics analysis shows that the near-grazing bifurcation will be unstable in three cases and a kind of co-dimension-two grazing bifurcation will generated in four cases.For this instability problem,a control strategy is proposed to realize the stability of the grazing bifurcation by controlling the existence of local attractors in the grazing periodic orbits.By means of discontinuous mapping method,the stability criterion of the double grazing bifurcation is obtained,under which a local attractor persists near a grazing trajectory.Then,based on the stability criterion,a two-parameter control region diagram is obtained,and a discrete-in-time feedback controller is designed on two Poincarésections to make local attractors exist near the grazing orbit,so as to achieve the control purpose.Besides,the chaos near co-dimension-two grazing bifurcation points was controlled by the control trategy.Finally,the feasibility of the control method is verified by numerical simulation.Secondly,the KM model for the dorsal cochlear nucleus(DCN)pyramidal neurons based on voltage-dependent ion channel variables and the simplified Integrate-and-Fire neuron model KM-LIF model are studied.The dynamical behaviors are explained by using bifurcation methods and fast/slow dynamical analysis when the fast potassium current deactivation variableh_f of the system was selected as bifurcation parameter.Then the model is controlled by the linear and nonlinear wash filter auxiliary dynamic feedback controller.For KM model,the Hopf bifurcation point position of KM model is controlled by linear filter.In addition,the oscillation of periodic motion after bifurcation is stabilized by nonlinear filter.For KM-LIF model,the bifurcation property of KM-LIF model is changed by linear filter,and the SN bifurcation is changed to Hopf bifurcation and controlled to a new position.The periodic motion oscillation after bifurcation is stabilized by nonlinear filter.