Renormalization Group Method For Perturbed Delay Differential Equations

In recent years,perturbed delay differential equations have been widely used in various fields,such as biology,fluid mechanics,engineering,economics,etc.,so they have been widely concerned by many scholars.Due to the emergence of time delay,which brings great difficulties to the exact solution of the equation.In order to obtain uniformly valid approximate solutions,perturbation methods such as the averaging method,matched method and multi-scale method have been developed successively,and a series of important research results have been obtained.In this paper,we apply the singularly perturbed renormalization group method to investigate the perturbed delay differential equations.The full text is divided into six chapters.Chapter 1 is the introduction,which gives a brief overview of the research background of delay differential equations and singular perturbation theory,and introduces the main work of this paper.In Chapter 2,the spectral decomposition theory and center manifold theorem of delay differential equations are introduced.In Chapter 3,we consider the initial value problems of single delay perturbed differential equations.Firstly,by means of singularly perturbed renormalization group method,an approximate solution of the perturbation problem is constructed.Secondly,we proved the uniform validness of the approximate solution.Finally,two examples are given to compare the results of this paper with the results of the averaging method.In Chapter 4,the initial value problem of perturbed differential equations with two delays is further considered.Firstly,for the case of two commensurate delays,we utilized the singularly perturbed renormalization group method to construct the approximate solution and proved the uniform validness of the approximate solution.Secondly,for the case of two incommensurate delays,the approximate solution of the original equation is constructed by introducing an approximate auxiliary system,and the error estimate is given.Finally,two examples are given to illustrate the validness of the proposed method,and the results are numerically compared with those obtained by the averaging method and the multi-scale method.In Chapter 5,firstly,for a class of weakly nonlinear perturbed delay differential equations,the approximate solution is obtained by singularly perturbed renormalization group method,and the uniform validness of the approximate solution is proved.Secondly,for a class of special weak nonlinear equations with delay in both linear and nonlinear terms,we combine spectral decomposition theory,center manifold theory and singularly perturbed renormalization group method to construct the approximate solution,and obtain the normal form of the reduced equation limited to the center manifold.Finally,we present a brief conclusion of the dissertation and the prospect of future work in Chapter 6.