by Chin Yu-Tung. / Thesis (M.Phil)--Chinese University of Hong Kong, 1981. / Bibliography: leaf 32.

This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations.

Typescript (photocopy). / Digitized by Kansas Correctional Industries / K-State Libraries' copy missing leaf 1 of introduction.

Dissertation (Ph.D.)--Auburn University,2007. / Abstract. Includes bibliographic references (p.99-103).

Thesis (PhD)--Montana State University--Bozeman, 2009. / Typescript. Chairperson, Graduate Committee: Tomas Gedeon. Includes bibliographical references (leaves 120-131).

Thesis (Ph.D.)--City University of Hong Kong, 2005. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy" Includes bibliographical references (leaves 121-133)

Delay differential equations (DDEs) are often used to model systems with time-delayed effects, and they have found applications in fields such as climate dynamics, biosciences, engineering, and control theory. In contrast to ordinary differential equations (ODEs), the phase space associated even with a scalar DDE is infinite-dimensional. Oftentimes, it is desirable to have low-dimensional ODE systems that capture qualitative features as well as approximate certain quantitative aspects of the DDE dynamics. In this thesis, we present a Galerkin scheme for a broad class of DDEs and derive convergence results for this scheme. In contrast to other Galerkin schemes devised in the DDE literature, the main new ingredient here is the use of the so called Koornwinder polynomials, which are orthogonal polynomials under an inner product with a point mass. A main advantage of using such polynomials is that they live in the domain of the underlying linear operator, which arguably simplifies the related numerical treatments. The obtained results generalize a previous work to the case of DDEs with multiply delays in the linear terms, either discrete or distributed, or both. We also consider the more challenging case of discrete delays in the nonlinearity and obtain a convergence result by assuming additional assumptions about the Galerkin approximations of the linearized systems. / Master of Science / Delay differential equations (DDEs) are equations that are commonly used to model systems with time-delayed effects. DDEs have found applications in fields such as climate dynamics, biosciences, engineering, and control theory. However, the solutions to these equations can be dicult to approximate. In a previous paper, a method to approximate certain types of DDEs was described. In this thesis, it is shown that this method may also approximate more general types of DDEs.

The stability and boundedness theories are developed for impulsive differential equations with time delay. Definitions, notations and fundamental theory are presented for delay differential systems with both fixed and state-dependent impulses. It is usually more difficult to investigate the qualitative properties of systems with state-dependent impulses since different solutions have different moments of impulses. In this thesis, the stability problems of nontrivial solutions of systems with state-dependent impulses are ``transferred" to those of the trivial solution of systems with fixed impulses by constructing the so-called ``reduced system". Therefore, it is enough to investigate the stability problems of systems with fixed impulses. The exponential stability problem is then discussed for the system with fixed impulses. A variety of stability criteria are obtained and`numerical examples are worked out to illustrate the results, which shows that impulses do contribute to the stabilization of some delay differential equations. To unify various stability concepts and to offer a general framework for the investigation of stability theory, the concept of stability in terms of two measures is introduced and then several stability criteria are developed for impulsive delay differential equations by both the single and multiple Lyapunov functions method. Furthermore, boundedness and periodicity results are discussed for impulsive differential systems with time delay. The Lyapunov-Razumikhin technique, the Lyapunov functional method, differential inequalities, the method of variation of parameters, and the partitioned matrix method are the main tools to obtain these results. Finally, the application of the stability theory to neural networks is presented. In applications, the impulses are considered as either means of impulsive control or perturbations.Sufficient conditions for stability and stabilization of neural networks are obtained.

No description available.

In this thesis we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.