Dynamics Of Lattice Differential Equations And State-dependent Delay Differential Equations

This dissertation is focused on the dynamics of lattice differential equations(LDEs) and differential equations with state-dependent delay(sd-DDEs). Usually,LDEs are infinite systems of ordinary differential equations indexed by points on a spatial lattice. With the advanced development of computers, numerical analysis becomes one of important background of LDEs. In addition, models involving LDEs can be found in many scientific disciplines, including image processing,chemical reaction theory, material science and population ecology etc. Compared with the continuous partial differential equations(PDEs), LDEs not only describe the actual problem more exactly, but also have been found to exhibit much more complicated dynamics. There are no many theoretical results on sd-DDEs. Since 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 theory and practice to study such equations. In this dissertation, we mainly investigate the existence of wave trains in a two-dimensional lattice and the dynamics of two classes of differential equations with state-dependent delays. This dissertation is organized as follows.Firstly, we study the existence and branching patterns of wave trains in a twodimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov-Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p : q resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tan θis rational.Secondly, we study the dynamics of the Nicholson blowflies equation with state-dependent delay. Under some suitable assumptions on the delay functions and parameters, we first study some basic properties including existence and continuous dependence on initial values of solutions of the system etc. Next, we construct a suitable compact state space such that solutions of the system constitute a continuous semiflow. By employing a discrete Lyapunov functional to study the slowly oscillating solutions, we show that all the globally defined slowly oscillating solutions form a global attractor of the semiflow. Thirdly, we consider the linearization and its spectrum and show the local dynamics of the system at the trivial solution, and obtain continuously differentiable local unstable manifold for the semiflow. At last, we choose a sufficiently small neighborhood of the positive equilibrium in the local unstable manifold and extend it to a global unstable manifold. By investigating the zero sets of solutions in the global unstable manifold,we prove the existence of the slowly oscillating periodic solution, which is exactly the boundary of the global unstable manifold.Finally, we study the existence of slowly oscillating periodic solutions in a second-order differential equation with state-dependent delay. At the beginning,we obtain some basic properties including existence and continuous dependence on initial values of solutions. Then we present a detailed analysis on the slowly oscillating solution. As a simple consequence of these consideration, we construct a compact set and a return map. Using the ejectivity of the trivial solution and the fixed point theorem, we prove the existence of the slowly oscillating periodic solution.