Dynamics Of State-dependent Delay Differential Equations

This dissertation is devoted to the dynamics of differential equations with state-dependent delay(sd-DDEs). So far, a perfect theoretical framework has not been established for state-dependent delay differential equations. Existing results about sd-DDEs are very few in both theory and applications. However, more and more sd-DDEs are used to describe the dynamical behaviors of models originated from physics, automatic control, neural networks, infectious diseases, population growth,and cell production etc. It is more meaningful and valuable in both theory and practice to study such equations.In this dissertation, we mainly investigate the dynamics of two classes of differential equations with state-dependent delays.This dissertation is organized as follows.Firstly, we study the dynamics of dynamics of van der Pol equations with three forms of state-dependent delay. The boundedness, uniqueness and continuous dependence of solutions on the initial value of the solution are firstly obtained by analyzing the structure of the first explicit van der Pol state-dependent delay differential equation. By constructing an appropriate compact set and a continuous compact mapping, according to the equilibrium state of the injectivity, using the fixed point theorem, we obtain the existence of the periodic solution of the slow oscillation. Then, we study the existence of slowly oscillating periodic solutions of the second van der Pol equations whose delay is determined by the state and the delay itself. The difference between this model and the first model is the delay function, which makes the Poincaréconstructed map discontinuous. In order to be able to make use of a suitable fixed point theorem, we look for the proper compact set, and unitize the time interval on it, then we construct an auxiliary compact state space and a Poincarémap on this compact state space. In order to solve the problem of the discontinuous delay term in the equilibrium, we construct a new Poincarémap. Now, the constructed system remains invariance in the auxiliary compact state space. We use the fixed point of quasi injectivity to study the local dynamical properties and prove the existence of the periodic solution of the slow oscillation. Finally, we give a full discussion about the dynamical behavior of the third state dependent delayed differential van der Pol equation. We study the local stability of the equilibrium, the asymptotics of the solutions, and the existence of the Hopf bifurcation. By the way of perturbation and the Fredholm orthogonality formula, we can determine whether the local bifurcation is supercritical Hopf bifurcation or subcritical. Also, we can obtain the stability of bifurcated periodic solution. The theoretical analysis is verified by the method of numerical simulation of three common delay functions.Secondly, we investigate the dynamics of prey-predator model. First of all,under appropriate assume of the several parameters, we study the local stability of the positive equilibrium, the oscillatory behavior of the solution, the parameter range where Hopf bifurcation occurrs. By a perturbation at the positive equilibrium, we reduce the Hopf bifurcation problem of the state-dependent delay model to that of the constant delay system. The local Hopf bifurcation theorem has been obtained by the Fredholm orthogonal formula. Then we determine the sub/supercritical Hopf bifurcation and hence the stability of the bifurcating periodic solutions. The theoretical analysis is verified by the method of numerical simulation of three common delay functions.