Yuri A. Mitropolsky – Författare

Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene- trated deep into various applied branches (theoretical physics, mechanics, ap- plied astronomy, dynamics of space flights, and others) and laid the founda- tion for lrumerous generalizations and for the creation of various modifications of thesem. E!f,hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay diffe,'ential equations and others). The stat. e of studying related problems was described in mono- graphs and original papers of Krylov N. M. , Bogoliubov N. N. [1], [2], Bogoli- ubov N. N [1J, Bogoliubov N. N. , Mitropolsky Yu. A. [1], Bogoliubov N. N. , Mitropol- sky Yu. A. , Samoilenko A. M. [1], Akulenko L. D. [1], van den Broek B. [1], van den Broek B. , Verhulst F. [1], Chernousko F. L. , Akulenko L. D. and Sokolov B. N. [1], Eckhause W. [l], Filatov A. N. [2], Filatov A. N. , Shershkov V. V. [1], Gi- acaglia G. E. O. [1], Grassman J. [1], Grebennikov E. A. [1], Grebennikov E. A., Mitropolsky Yu. A. [1], Grebennikov E. A. , Ryabov Yu. A. [1], Hale J . K. [I]' Ha- paev N. N. [1], Landa P. S. [1), Lomov S. A. [1], Lopatin A. K. [22]-[24], Lykova O. B.

This volume is devoted to the further development of the asymptotic theory for analyzing solutions of a range of nonlinear periodic boundary value problems. It suggests a systematic approach to constructing asymptotic methods for solving wave equations, hyperbolic differential equations and partial differential equations with small parameters. The book should be of interest to researchers and postgraduate students whose work involves partial differential equations, mathematical physics, or approximations and expansions.

This volume addresses the application of asymptotic methods in nonlinear oscillations. Such methods see a large variety of applications in physics, mechanics and engineering. The advantages of using asymptotic methods in solving nonlinear problems are firstly simplicity, especially for computing higher approximations, and secondly their applicability to a large class of quasi-linear systems. The book is concerned with the applied aspects of the methods concerned and also contains problems relevant to the everyday practice of engineers, physicists and mathematicians. Usually, dynamics systems are classified and examined by their degrees of freedom. This text is constructed from another point of view based upon the originating mechanism of the oscillations: free oscillation, self-excited oscillation, forced oscillation, and parametrically excited oscillation. The text has been designed to cover material from the elementary to the more advanced, in increasing order of difficulty. It should be of intereest to both students and researchers in applied mathematics, physical and mechanical sciences, and engineering.

Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene- trated deep into various applied branches (theoretical physics, mechanics, ap- plied astronomy, dynamics of space flights, and others) and laid the founda- tion for lrumerous generalizations and for the creation of various modifications of thesem. E!f,hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay diffe,'ential equations and others). The stat. e of studying related problems was described in mono- graphs and original papers of Krylov N. M. , Bogoliubov N. N. [1], [2], Bogoli- ubov N. N [1J, Bogoliubov N. N. , Mitropolsky Yu. A. [1], Bogoliubov N. N. , Mitropol- sky Yu. A. , Samoilenko A. M. [1], Akulenko L. D. [1], van den Broek B. [1], van den Broek B. , Verhulst F. [1], Chernousko F. L. , Akulenko L. D. and Sokolov B. N. [1], Eckhause W. [l], Filatov A. N. [2], Filatov A. N. , Shershkov V. V. [1], Gi- acaglia G. E. O. [1], Grassman J. [1], Grebennikov E. A. [1], Grebennikov E. A., Mitropolsky Yu. A. [1], Grebennikov E. A. , Ryabov Yu. A. [1], Hale J . K. [I]' Ha- paev N. N. [1], Landa P. S. [1), Lomov S. A. [1], Lopatin A. K. [22]-[24], Lykova O. B.

The present volume addresses the application of asymptotic methods in nonlinear oscillations. Such methods see a large variety of applications in physics, mechanics and engineering. The advantages of using asymptotic methods in solving nonlinear problems are firstly simplicity, especially for computing higher approximations, and secondly their applicability to a large class of quasi-linear systems. In contrast to the existing literature, this book is concerned with the applied aspects of the methods concerned and also contains problems relevant to the everyday practice of engineers, physicists and mathematicians. Usually, dynamics systems are classified and examined by their degrees of freedom. This book is constructed from another point of view based upon the originating mechanism of the oscillations: free oscillation, self-excited oscillation, forced oscillation, and parametrically excited oscillation. The text has been designed to cover material from the elementary to the more advanced, in increasing order of difficulty. It is of considerable interest to both students and researchers in applied mathematics, physical and mechanical sciences, and engineering.

Many problems in celestial mechanics, physics and engineering involve the study of oscillating systems governed by nonlinear ordinary differential equations or partial differential equations. This volume represents an important contribution to the available methods of solution for such systems. The contents are divided into six chapters. Chapter 1 presents a study of periodic solutions for nonlinear systems of evolution equations including differential equations with lag, systems of neutral type, various classes of nonlinear systems of integro-differential equations, etc. A numerical-analytic method for the investigation of periodic solutions of these evolution equations is presented. In Chapters 2 and 3, problems concerning the existence of periodic and quasiperiodic solutions for systems with lag are examined. For a nonlinear system with quasiperiodic coefficients and lag, the conditions under which quasiperiodic solutions exist are established. Chapter 4 is devoted to the study of invariant toroidal manifolds for various classes of systems of differential equations with quasiperiodic coefficients. Chapter 5 examines the problem concerning the reducibility of a linear system of difference equations with quasiperiodic coefficients to a linear system of difference equations with constant coefficients. Chapter 6 contains an investigation of invariant toroidal sets for systems of difference equations with quasiperiodic coefficients. For mathematicians whose work involves the study of oscillating systems.