Research On Dynamics Analysis And Control For Two Classes Of Nonlinear Systems With Special Structures

Dynamic systems with special nonlinear structures have wide-ranging practical application backgrounds,and there are many complex nonlinear phenomena.Research on their dynamic characteristics and production mechanism has become one of the hot topics in the field of dynamics and control.Because of the strong nonlinearity and singularity,the system with special structure will possess some special dynamic characteristics,which cannot be solved by the traditional nonlinear theory.Therefore,it is necessary to further develop the corresponding theory system.In this dissertation,we investigate the smooth dynamic systems with time delay,and the non-smooth dynamic system with multi-scale factor,respectively.By employing the stability theory of delay differential equations,bifurcation theory,control theory as well as the differential inclusion theory of the Filippov system,etc.,the complex dynamic behaviors and the mechanism of these two types of systems are exploredThe main work of this dissertation includes the following aspects1.The dynamic properties for a nonlinear system with two delays are studied.By selecting the predator-prey model,a smooth dynamical system including two delays and a prey refuge where the predator population consumes the prey in accordance with Beddington-DeAngelis type functional response is established.Firstly,we investigate the positivity,boundedness,existence of equilibrium points and the local stability of the each of feasible equilibrium points in the system without delay.Secondly,with the help of two methods,namely the comparison theorem as well as constructing an appropriate Lyapunov function and combining the LaSalle invariance principle,the related conditions for the global stability of the equilibrium points of non-delayed system are derived,respectively.Thirdly,choosing both time delays as the bifurcation parameter,the local stability,existence of Hopf bifurcation and global stability of delayed system are discussed,and on the basis of dimension reduction idea of center manifold theorem,the nature of Hopf bifurcation is also acquired.Finally,representative numerical simulations are provided to illustrate the theoretical analysis,and the biological explanations of the effects of two special nonlinear structures on system dynamics,namely the time delay and the prey refuge,are given2.The bifurcation and control problem for a delayed stage-structured nonlinear system are explored.Firstly,we discuss the positivity and the existence of equilibrium points.Then,by choosing time delay as the bifurcation parameter and analyzing the relevant characteristic equations,the local stability of the trivial equilibrium,the boundary equilibrium as well as the positive equilibrium of the system are investigated And the direction of Hopf bifurcation and stability of the bifurcating periodic solution at the positive equilibrium point are obtained.Furthermore,for the purpose of protecting the stability of such biological system,a hybrid control method based on state feedback and parameter regulation is presented to control the Hopf bifurcation.Finally,numerical examples are given to verify the theoretical findings3.The bifurcation control of a nonlinear system with two delays is studied by using a hybrid control strategy which is based on state feedback and parameter regulation.For a predator-prey model as an example,by analyzing the associated characteristic equation,its local stability and the existence of Hopf bifurcation with respect to both delays are analyzed.It is found that the control method delays the inherent bifurcation of the original system.Based on the normal form theory and the center manifold theorem,explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution.Numerical simulation results confirm that the hybrid controller is efficient and feasible in controlling Hopf bifurcation4.The mixed-mode oscillations and bifurcation mechanism for a Filippov-type system with two scales in the frequency domain are revealed.Based on the classic Chua's circuit system,a non-smooth model with two time scales is established.Via the stability analysis of equilibrium points,the critical conditions of the fold bifurcation and Hopf bifurcation are given.By employing the differential inclusions theory,an auxiliary parameter is introduced,which can be applied to explore the non-smooth bifurcations when the trajectory of system passes across the non-smooth boundary.Numerical simulations with different parameter are presented,corresponding to different types of bursting oscillations.From the bifurcation conditions,and through overlapping the equilibrium branches with the transformed phase portrait,the mechanism of the transitions of different states in periodic bursting oscillations are investigated.