Stability And Bifurcation On Some Delay Differential Equations

Delay differential equations are used to describe evolution systems that depend on both the current state and the past state.In the fields of biological mathematics,physics and economics,research on the dynamical behavior(such as the stability of equilibrium,bifurcation and chaos,etc.)of delay differential equation models with practical application background becomes a hot topic in recent decades.Hence,it is of great theoretical and practical significance to study the long time dynamical properties of systems with different delays.This paper based on the stability theory,bifurcation theory of nonlinear dynamical systems,combined with the central manifold theorem,normal form method,first approximation method and Lyapunov’s second method and other mathematical methods,the dynamical behaviors of predator prey model with reserved area,Shimizu-Morioka model,competition and cooperation system with constant delays are studied in detail.This paper is divided into four chapters.The second,third and fourth chapters are main work.1.The first chapter is the introduction which summarizes the development and the Hopf bifurcation theory of functional differential equations,and expounds the main work of this paper.2.The second chapter studied the effect of delay on the dynamical behaviors of a three-species predator-prey model with a protected area.Without delay term,the positive equilibrium point is globally asymptotic stability.For the model with delay term,we chose delay as the bifurcation parameter,the existence of Hopf bifurcation and the stability of periodic solutions of Hopf bifurcation are analyzed.Meanwhile,XPPAUT and DDE-BIFTOOL package in Matlab are used to explore the chaos of delayed system.Finally,the biological explanation of how delay affects the dynamical properties of the system are given.3.The third chapter studies the Shimizu-Morioka model with constant delay.Firstly,the existence of the equilibrium is determined,and the local stability of the equilibrium is explained by using the characteristic equation method.Secondly,taking time delay as the bifurcation parameter,the Hopf bifurcation is analyzed by using the Hopf bifurcation theory of functional differential equations.Finally,numerical simulations are used to verify the correctness of the theoretical analysis results.4.Chapter four discusses the existence and stability of the positive equilibrium for the discrete delay competitive cooperation model.Taking time delay as the bifurcation parameter,the detailed calculation formula determining Hopf bifurcation direction and stability of periodic solution is given by using the center manifold theory and normal form method.Ode45,dde23 and other function handles of Matlab are applied in numerical simulation for theoretical analysis results.Finally,the economic significance of the original model is explained to provide theoretical support for practical problems.