Impulsive Differential Equations: Periodic Solutions and Applications PDF

Preface

Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is, jumps) in their values. Processes of this type are often investigated in various fields of science and technology.

Particular impulsive differential equations simulating the work of concrete systems have been considered by many authors. As an example of this we shall point out the investigations of the shock model of a clock mechanism (Andronov et al. [6], Bautin [17], Kalitin [47-49], and Krylov and Bogolyubov [53]) and the investigations of the relaxation oscillations of electromechanical systems (Andronov et al. [6]).

The mathematical theory of systems that change by jumps is developing in two main directions.

One of these directions is delineated in the works on impulsive systems by Halanay and Wexler [34], Pandit and Deo [61], and Zavaliscin et al. [91]. In these publications the apparatus of generalized functions is used and differential equations with fixed moments of the impulse effect are investigated regardless of the fact that the impulse effects can occur at moments when certain spatial or space-time relations are valid.

Investigations in the other direction were begun by Mil’man and Myshkis [57]. They give some general concepts of systems with an impulse effect and obtain the first results on the stability of their solutions. Their results make it possible to investigate impulsive systems for which the jumps take place when certain space-time relations are valid. The well-known classical apparatus of ordinary differential equations is used. The authors of the present book adhere to this direction.

The possibility of wide practical application of impulsive differential equations explains the still growing interest of many authors in the investigation of these equations in recent years and the publication of monographs on this subject by A. M. Samoilenko and N. A. Perestyuk [73], V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov [55], and D. D. Bainov and P. S. Simeonov [15].

One of the important trends in the investigation of impulsive differential equations is related to the periodic solutions of these equations. To this subject many of the results obtained by M. U. Ahmetov, D. D. Bainov, V. I. Gutu, S. G. Hristova, V. Lakshmikantham, Xinzhi Liu, N. A. Perestyuk, A. M. Samoilenko, and P. S. Simeonov are devoted.

In the present book a systematic exposition of the results related to periodic solutions of impulsive differential equations is given and the potential for their application is illustrated.

The book consists of six chapters. Chapter I is introductory and offers a description of impulsive differential equations, as well as a brief exposition of auxiliary assertions about these equations which are used in subsequent chapters.

In Chapter II linear impulsive differential equations with fixed moments of an impulse effect are considered. The Floquet theory for these equations is expounded, the problem of the existence of periodic solutions of non-homogeneous linear equations in the non-critical and the critical cases is solved, and Hamiltonian periodic impulsive equations are considered.

In Chapters III and IV the method of the small parameter is elaborated for the three main classes of impulsive equations considered in the book. In Chapter III the non-critical case is considered, and Chapter IV deals with the critical case.

In Chapter V the problem of the existence of periodic solutions of non-linear impulsive equations is discussed, applying the method of the monodromy operator along trajectories (the Poincare operator) and the method of upper and lower solutions.

In Chapter VI various approximate methods for finding periodic solutions of impulsive differential equations are considered, such as the monotone-iterative technique, a numerical-analytical method, the method of bilateral approximations, a projectioniterative method, and the method of boundary layer functions for singularly perturbed impulsive equations.

Possible practical applications of the results obtained are illustrated by a large number of concrete mathematical models, and an extensive bibliography on the subjects treated in the book is given.

The authors wish to express their gratitude to the staff at Longman and to Dr Alan Jeffrey for their support in publishing the book. They also express sincere thanks to Dr V. Covachev who helped in the preparation of the manuscript of the book, and to the Bulgarian Ministry of Education and Science for their partial support under Grant MM-7. 

Authors: Drumi Bainov, Pavel Simeonov