Asymptotic Methods in Resonance Analytical Dynamics

Stability and Control: Theory, Methods and Applications Volume 21

Asymptotic Methods in Resonance Analytical Dynamics E. A. Grebenikov Russian Academy of Sciences Moscow, Russia

Yu. A. Mitropolsky National Academy of Sciences of Ukraine Kiev, Ukraine

Yu. A. Ryabov Moscow Institute of Auto and Highway Construction Moscow, Russia

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

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Library of Congress Cataloging-in-Publication Data Grebenikov, E. A. (Evgeniæi Aleksandrovich) Asymptotic methods in resonance analytical dynamics / by Eugeniu Grebenikov, Yu. A. Mitropolsky, and Y. Ryabov. p. cm. — (Stability and control ; v. 21) Includes bibliographical references and index. ISBN 0-415-31008-3 (alk. paper) 1. Averaging method (Differential equations) 2. Differential equations—Asymptotic theory. 3. Resonance—Mathematical models. I. Mitropol§skiæl, ëIìU. A. (ëIìUriæi Alekseevich), 1917- II. Rëiìabov, ëIìU. A. (ëIìUriæi Aleksandrovich) III. Title. IV. Series. QA372 .G715 2004 515¢.35—dc22

2003069583

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Visit the CRC Press Web site at www.crcpress.com © 2004 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-415-31008-3 Library of Congress Card Number 2003069583 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

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Introduction to the Series The problems of modern society are both complex and interdisciplinary. Despite the apparent diversity of problems, tools developed in one context are often adaptable to an entirely different situation. For example, consider the Lyapunov’s well known second method. This interesting and fruitful technique has gained increasing signifycance and has given a decisive impetus for modern development of the stability theory of differential equations. A manifest advantage of this method is that it does not demand the knowledge of solutions and therefore has great power in application. It is now well recognized that the concept of Lyapunov-like functions and the theory of differential and integral inequalities can be utilized to investigate qualitative and quantitative properties of nonlinear dynamic systems. Lyapunov-like functions serve as vehicles to transform the given complicated dynamic systems into a relatively simpler system and therefore it is sufficient to study the properties of this simpler dynamic system. It is also being realized that the same versatile tools can be adapted to discuss entirely different nonlinear systems, and that other tools, such as the variation of parameters and the method of upper and lower solutions provide equally effective methods to deal with problems of a similar nature. Moreover, interesting new ideas have been introduced which would seem to hold great potential. Control theory, on the other hand, is that branch of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems. To control an object implies the influence of its behavior so as to accomplish a desired goal. In order to implement this influence, practitioners build devices that incorporate various mathematical techniques. The study of these devices and their interaction with the object being controlled is the subject of control theory. There have been, roughly speaking, two main lines of work in control theory which are complementary. One is based on the idea that a good model of the object to be controlled is available and that we wish to optimize its behavior, and the other is based on the constraints imposed by uncertainty about the model in which the object operates. The control tool in the latter is the use of feedback in order to correct for deviations from the desired behavior. Mathematically, stability theory, dynamic systems and functional analysis have had a strong influence on this approach. Volume 1, Theory of Integro-Differential Equations, is a joint contribution by V. Lakshmikantham (USA) and M. Rama Mohana Rao (India). Volume 2, Stability Analysis: Nonlinear Mechanics Equations, is by A. A. Martynyuk (Ukraine). Volume 3, Stability of Motion of Nonautonomous Systems: The Method of Limiting Equations, is a collaborative work by J. Kato (Japan), A. A. Martynyuk (Ukraine) and A. A. Shestakov (Russia).

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Volume 4, Control Theory and Its Applications, is by E. O. Roxin (USA). Volume 5, Advances in Nonlinear Dynamics, is edited by S. Sivasundaram (USA) and A. A. Martynyuk (Ukraine) and is a multiauthor volume dedicated to Professor S. Leela (USA). Volume 6, Solving Differential Problems by Multistep Initial and Boundary Value Methods, is a joint contribution by L. Brugnano (Italy) and D. Trigiante (Italy). Volume 7, Dynamics of Machines with Variable Mass, is by L. Cveticanin (Yugoslavia). Volume 8, Optimization of Linear Control Systems: Analytical Methods and Computational Algorithms, is a joint contribution by F. A. Aliev (Azerbaijan) and V. B. Larin (Ukraine). Volume 9, Dynamics and Control, is edited by G. Leitmann (USA), F. E. Udwadia (USA) and A. V. Kryazhimskii (Russia) and is a multiauthor volume. Volume 10, Volterra Equations and Applications, is edited by C. Corduneanu (USA) and J. W. Sandberg (USA) and is a multiauthor volume. Volume 11, Nonlinear Problems in Aviation and Aerospace, is edited by S. Sivasundaram (USA) and is a multiauthor volume. Volume 12, Stabilization of Programmed Motion, is by E. Ya. Smirnov (Russia). Volume 13, Advances in Stability Theory at the End of the 20th Century, is edited by A. A. Martynyuk (Ukraine) and is a multiauthor volume. Volume 14, Dichotomies and Stability in Nonautonomous Linear Systems, is a collaborative work by Yu. A. Mitropolsky (Ukraine), A. M. Samoilenko (Ukraine) and V. L. Kulik (Ukraine). Volume 15, Almost Periodic Solutions of Differential Equations in Banach Spaces, is a collaborative work by Y. Hino (Japan), T. Naito (Japan), Nguyen Van Minh (Vietnam) and Jong Son Shin (Japan). Volume 16, Functional Equations with Causal Operators, is by C. Corduneanu (USA). Volume 17, Optimal Control of the Growth of Wealth of Nations, is by E. N. Chukwu (USA). Volume 18, Stability and Stabilization of Nonlinear Systems with Random Structure, is a joint contribution by I. Ya. Kats (Russia) and A. A. Martynyuk (Ukraine). Volume 19, Lyapunov Functions in Differential Games, is by V. I. Zhukovskiy (Russia). Volume 20, Stability of Differential Equations with Aftereffect, is a joint contribution by N. V. Azbelev (Russia) and P. M. Simonov (Russia). Volume 21, Asymptotic Methods in Resonance Analytical Dynamics, is e a collaborative work by E. A. Grebenikov (Russia), Yu. A. Mitropolsky (Ukraine) and Yu. A. Ryabov (Russia). Due to the increased interdependency and cooperation among the mathematical sciences across the traditional boundaries, and the accomplishments thus far achieved in the areas of stability and control, there is every reason to believe that many breakthroughs await us, offering existing prospects for these versatile techniques to advance further. It is in this spirit that we see the importance of the ‘Stability and Control’ series, and we are immensely thankful to Taylor & Francis and Chapman & Hall/CRC for their interest and cooperation in publishing this series.

Copyright © 2004 CRC Press LLC


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Contents Introduction to the Series Preface Acknowledgments 1 Preliminaries 1.0 Introduction 1.1 Main Symbols 1.2 Asymptotic Series and their Properties 1.3 Poincar´e’s Theorem on Asymptotic Approximations of Solutions of Differential Equations 1.4 Geometric Interpretation of Solutions of Oscillating Systems 1.5 On the Method of Characteristics for Quasi-Linear First-Order Partial Differential Equations: Method of Characteristics 1.6 Iterative Variant of the Poincar´e–Lyapunov Small Parameter Method 1.6.1 Simple iterations 1.6.2 Iterations with quadratic convergence 1.7 Comments and References

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2 Averaging Principle for Multifrequency Systems of Differential Equations 2.0 Introduction 2.1 Mathematical Averaging Principle 2.2 Classification of Systems of Differential Equations where Resonances are Possible 2.3 The Basis of the Asymptotic Theory for Locally Nonresonance Systems 2.4 Initial Conditions for Comparison Equations 2.5 Averaging Operator for Time-Independent Disturbances 2.6 Asymptotic Theory of Systems with Their Paths Passing through Resonance Points 2.7 The Algorithm of Joining of Resonance and Nonresonance Path Sections 2.8 Periodic and Quasi-Periodic Oscillations in the Van der Pol Oscillator System 2.9 Study of Multifrequency Systems with Their Solutions Not Remaining Close to Resonance Points 2.10 Study of Multifrequency Systems Belonging to Class II 2.11 Multifrequency Systems with Their Solutions Not Leaving the Neighborhood of a Resonance Point 2.12 Comments and References 3 Some Resonance Problems of Nonlinear Mechanics 3.0 Introduction 3.1 Newtonian Three-Body Problem 3.2 The Problem of Justification of the Averaging Principle in the Bounded Newtonian Three-Body Problem 3.3 Construction of Explicit Solutions of Averaged Differential Equations of the Bounded Three-Body Problem in the Case of Resonance 3.4 Quasi-Periodic Solutions of Resonance Hamiltonian Systems

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3.5 Motion of a Geostationary Satellite 3.6 Averaging Method in the Theory of Partial Differential Equations 3.7 Energy Method of Construction of Amplitude–Phase Equations 3.8 Averaging Method and Maximum Principle in Boundary Value Problems 3.9 Comments and References 4 Numerical–Analytic Methods 4.0 Introduction 4.1 Construction of Lyapunov Transform for a Linear System with Periodic Coefficients 4.1.1 Construction of matrices L(t) and W by means of series 4.1.2 Construction of matrices L(t) and W by means of iterations 4.1.3 Interpolation formulae for Lyapunov transform matrices 4.2 Construction of Green and Lyapunov Matrices 4.2.1 Noncritical case 4.2.2 Example 4.2.3 Critical cases 4.3 Direct Numerical–Analytic Method of Construction of Periodic Solutions 4.4 Construction of Periodic Solutions in Hill’s Problem of Lunar Motion 4.5 Numerical–Analytic Construction of Mathieu Functions 4.5.1 Algorithm construction 4.5.2 Computational layout 4.5.3 Quick-Basic program 4.5.4 Comments on the program 4.6 Algorithm for Construction of Solutions of the Plane Bounded Three-Body Problem 4.6.1 Initial differential equations of the problem 4.6.2 Basic equations for the coefficients of the sought solution 4.6.3 Construction of a solution by the method of simple iterations

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4.6.4 Construction of a solution by the method of iterations with quadratic convergence 4.7 Numerical–Analytic Implementation of Krylov–Bogolyubov Transform 4.8 Comments and References References

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