Asymptotic Methods for Ordinary Differential Equations [Hardcover]

In this book we consider a Cauchy problem for a system of ordinary differential equations with a small parameter. The book is divided into th ree parts according to three ways of involving the small parameter in the system. In Part 1 we study the quasiregular Cauchy problem. Th at is, a problem with the singularity included in a bounded function j , which depends on time and a small parameter. This problem is a generalization of the regu? larly perturbed Cauchy problem studied by Poincare [35]. Some differential equations which are solved by the averaging method can be reduced to a quasiregular Cauchy problem. As an example, in Chapter 2 we consider the van der Pol problem. In Part 2 we study the Tikhonov problem. This is, a Cauchy problem for a system of ordinary differential equations where the coefficients by the derivatives are integer degrees of a small parameter.In this book we consider a Cauchy problem for a system of ordinary differential equations with a small parameter. The book is divided into th ree parts according to three ways of involving the small parameter in the system. In Part 1 we study the quasiregular Cauchy problem. Th at is, a problem with the singularity included in a bounded function j , which depends on time and a small parameter. This problem is a generalization of the regu? larly perturbed Cauchy problem studied by Poincare [35]. Some differential equations which are solved by the averaging method can be reduced to a quasiregular Cauchy problem. As an example, in Chapter 2 we consider the van der Pol problem. In Part 2 we study the Tikhonov problem. This is, a Cauchy problem for a system of ordinary differential equations where the coefficients by the derivatives are integer degrees of a small parameter.Preface. Part 1: The Quasiregular Cauchy Problem. 1. Solutions Expansions of the Quasiregular Cauchy Problem. 2. The Van der Pol Problem. Part 2: The Tikhonov Problem. 3. The Boundary Functions Method. 4. Proof of Theorems 28.1-28.4. l£®