An Analysis Of Delay-Dependent Stability For Second Order Delay Differential Equations

Delay differential equations(DDEs) play an important role in describing the phenomenon in the fields of Physics, Engineering, Biology, Medical Science, Economics and so on. The numerical methods for solving DDEs is also valuable to concern with. For this reason, many papers have focused on this topic and a large number of important results have been found since 1960's. In this paper, we discuss the stability and delay-dependent stability region, both of analytical and numerical, of second order DDEs.At the beginning of this paper, we present many applications of delay differential equations in different fields and give a brief introduction to the development of the stability theory of DDEs in the past few decades.Secondly, we deal with the asymptotic stability of a class of second order DDEs with two real coefficients in the second chapter of this thesis. By citing the work from the literature, we plot the analytical stability region in the parameter plane. Next, the delay-dependent stability region of the trapezium rule is derived and its boundary is found. Then a comparison between analytical and numerical stability regions is made and it is proved that the analytical stability region is a subset of the numerical stability region, which shows that the trapezium rule completely preserves the asymptotic stability of the analytical solution of the model equation.In the third Chapter, we attempt to investigate a class of second order DDEs, which contains three real coefficients. With the study of the critical roots locus curve in the parameter plane, the exact stability region of continuous problem is obtained. Furthermore, a necessary and sufficient conditions is gained.Next, we consider the adaptation of the trapezium rule and study the numerical stability of the modal equation discussed in the last chapter. Following the same train of thought, we can get the numerical stability region too. And it is proved that the analytical stability region is a subset of the numerical stability region, which shows that the trapezium rule completely preserves the asymptotic stability of the analytical solution of this model equation.Finally, a numerical experiment is made to support our results in this paper.