The Complex Dynamics Of Two Kinds Of Differential Equations With State Dependent Impulsive Control

In recent years, the researches on IDE are transforming from the system with pulses at fixed times to it with state dependent impulsive effects. In this context, we mainly study the impulsive state feedback control of a Michaelis-Menten type predator-prey model and an SIR epidemic model with state dependent pulse vaccination in the paper. The main work is as follows:Firstly, the dynamics of a Michaelis-Menten type predator-prey model with state feedback control is studied to show the state change of the population.(1) We obtain the sufficient conditions for the existence and stability of semi-trivial solution and the existence of order-1positive periodic solution by using the Poincare Map.(2) The theoretical analysis reveals that the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation and the sufficient conditions for the stability of positive periodic solution is obtained by using the Analogue of the Poincare criterion.(3) We make a classified discussion of the positive periodic solution and confirm that the order-1periodic solution change into order-2periodic solution by Flip bifurcation and then leads to chaos.(4) Detailed numerical results of phase portraits, Lyapunov exponents and the bifurcation diagrams illustrated by an example are carried out to explicate the feasibility of our theoretical results.Secondly, on the base of deterministic SIR epidemic model with impulsive vaccination, a kind of SIR epidemic model with state dependent pulse vaccination is discussed by means of both theoretical and numerical analysis, whose infection rate equals to kIS. We firstly prove the order-1periodic solution by using the Poincare Map, then we find the conditions on which the periodic solution is orbitally asymptotically stable by using the Analogue of the Poincare criterion. Finally, Numerical analysis indicate the complex dynamics of the model which depends on the pulse vaccination parameter. The system experienced periodic solution varied period, strange attractor, chaos and a cascade of period-doubling bifurcations, which are in good agreement with the theoretical analysis. In a word, state dependent pulse vaccination strategy is a very economical, of high efficiency and easily-operational method to prevent various viral infectious diseases.