Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients

290 Pages·1993·19.74 MB·English

by  Yu A. Mitropolsky, A. M. Samoilenko, D. I. Martinyuk (auth.)

Systems of Evolution Equations
with Periodic and Quasiperiodic Coefficients
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, Russia
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, Russia
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, Russia
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, Russia
Yu. A. ROZANOV, Steklov Institute ofM athematics, Moscow, Russia
Volume 87
Systems of
Evolution Equations
with Periodic and
Quasiperiodic Coefficients
by
Yu. A. Mitropolsky,
A. M. Samoilenko
and
D.I. Martinyuk
Institute of Mathematics,
Kiev, Ukraine, Cl.S.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library ofCongress Cataloging-in-Publication Data
Mltropo 1 'sk 11. ro. A. (fiJr 11 Al ekssev 1e h). 1917-
systems of evolution equations with periodic and quasiperiodic
coefficients I by Yu. A. Mitropolsky. A.M. Samoilenko. and
D.I. Martinyuk.
p. eN. -- (Mathe.atles and lts applleatlons. Sovlet serles ;
87)
Ineludes Index.
ISBN 978-94-010-5210- 8 ISBN 978-94-011-2728-8 (eBook)
DOI 10.1007/978-94-011-2728-8
1. Evolutlon equatlons. 1. Sa.ollenko. A. M. (Anatoll1
Mlkhallovleh) II. Martlnyuk. D. 1. III. Tltla. IV. Serles:
Mathe.atles and lts appl1eatlons (Kluwer"Aeademle Publ1shers).
Sovlet sar les ; 87.
0AS71.M625 1992
515·.353--de20 92-37385
ISBN 978-94-010-5210-8
This is a translation of the original work
Systems 01 Evolution Equations with
Periodic and Conditionolly Periodic CoejJicients
Published by Nauka, Moscow, el 1984
AII Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint of the hardcover 1s t edition 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
'Et moi, .. ~ si favait su comment en revemr, One service mathematia bas rcndcmI the
je n'y scrais point aIle.' human m:e. It bas put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labdled 'diseardI:d non-
The series is divergent; therefore we may be sense'.
able to do something with it Eric T. Bell
o. Hcaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non-
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com-
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'ctre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly speciaJjzed topics. However, the 'tree' of knowledge of
mathematics and related fields d.oes not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdiscipIines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more e1fort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of interac-
tion areas one should add string theory where Riemann surfaces, algebraic geometry, modular func-
tions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come
together. And to the examples of things which can be usefully applied let me add the topic 'finite
geometry'; a combination of words which sounds like it might not even exist, let alone be applica-
ble. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite
projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no
part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accord-
ingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics,
the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on.
vi
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
extra mathematical sophistication that this requires. For that is where the rewards are. linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non-
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci-
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com-
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre-
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub-
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It bas been enlarged a bit to include books treating of the tools from one subdis-
cipline which are used in others. Thus the series still aims at books dealing with:
a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
new applications of the results and ideas from one area of scientific endeavour into another;
influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my Iibnuy
domain. an: books that other folk have lent me.
J. Hadamard Anatole France
La physique nc nous donne pas scu1emcnt The function of an expert is not to be more
I'occasion de ~udn: des problemcs ... cDc right than other people, but to be wrong for
nous fait presscntir Ia solution. more sophisticated reasons.
H. Poincare David Butler
Bussum Micbiel Hazewinkel
CONTENTS
Preface xi
CHAPTER 1. NUMERICAL-ANALYTIC METHOD OF
INVESTIGATION (}FPBRIODIC SOLtJTIONS
FOR SYSTEMS WITH AFTEREFFECT 1
§1. Notations and Basic Ideas. Auxiliary Lemmas 2
§2. Algorithm for Finding Periodic Solutions of Nonlinear Systems with Lag 6
§3. Existence of Periodic Solutions 13
§4. Construction of an Approximate Periodic Solution 18
§5. Periodic Problem of Control for Systems with Lag 26
§6. The Concrete Example of Application of the Numerical-Analytic Method 27
§7. Periodic Solutions of Differential Equations of the Second Order with
Retarded Argument 30
§8. Periodic Solutions for Countable Systems of Differential Equations
with Lag 38
§9. Periodic Solutions for Nonlinear Systems of Differential Equations
of the Neutral Type 42
§1O. Investigation of Periodic Solutions for Some Classes of Systems of
Integro-Differential Equations 45
§11. Periodic Solutions of Nonlinear Systems of Difference Equations 55
§12. Bilateral Approximations to Periodic Solutions of Systems with Lag 59
§13. Periodic Solutions of Operator-Differential Equations 71
CHAPTER 2. INVESTIGATION OF PERIODIC SOLUTIONS OF
SYSTEMS WITH AFTEREFFECT BY BUBNOV-
GALERKIN'S METHOD 77
§1. Preliminary Remarks. Auxiliary Statements 77
§2. Green's Function for the Problem of Periodic Solutions of Linear
Systems with Lag. Properties of This Function 83
§3. The Main Properties of the Jacobian Matrix for Determining
Equations for Galerkin' s Approximations 87
vii
viii Contents
§4. Existence and Convergence of Bubnov-Galerkin's Approximations 93
§5. Existence of Periodic Solutions for Systems of Differential Equations
with Lag 98
§6. Applications of Bubnov-Galerkin's Method to the Investigation
of Periodic Solutions for Some Classes of Systems of Integro-
Differential Equations 101
CHAPTER 3. QUASIPERIODIC SOLUTIONS OF SYSTEMS
WITH LAG. BUBNOV- GALERKIN'S METHOD 107
§ 1. Definitions and Auxiliary Statements 107
§2. Construction of Quasiperiodic Solutions of Systems with Lag by
Bubnov-Galerkin's Method 114
§3. Construction of Quasiperiodic Solutions of Perturbed Systems with
Lag by Bubnov-Galerkin's Method 123
CHAPTER 4. EXISTENCE OF INVARIANT TOROIDAL MANI-
FOLDS FOR SYSTEMS WITH LAG. INVESTIGA-
TION OF THE BEHAVIOR OF TRAJECTORIES
IN THEIR VICINITIES 135
§1. Existence ofInvariant Toroidal Manifolds with Loss of Smoothness 135
§2. Existence of Lipschitz Tori for Nonlinear Systems with Lag 150
§3. Invariant Toroidal Sets for the Systems of Differential Equations
with Lag Under Pulse Influence 169
§4. Behavior of Solutions of Nonlinear Systems with Lag in the Vicinity
of Exponentially Stable Toroidal Manifolds 185
CHAPTER 5.REDUCmILITY OF LINEAR SYSTEMS OF
DIFFERENCE EQUATIONS WITH QUASI-
PERIODIC COEFFICIENTS 201
§ 1. Statement of the Problem and Auxiliary Statements 201
§2. Theorem on Reducibility 206
§3. Reducibility of Linear Systems of Difference Equations with a
Smooth Right-Hand Side 214
Contents ix
CHAPTER 6.INVARIANT TOROIDAL SETS FOR SYSTEMS OF
DIFFERENCE EQUATIONS. INVESTIGATION OF
THE DEHAV IOR OF TRAJECTORIES ON TOROI-
DAL SETS AND IN THEIR VICINITIES 223
§ 1. Auxiliary Lemmas 223
§2. Periodic Solutions to Some Classes of Functional Equations 229
§3. Existence of Invariant Toroidal Sets for Nonlinear Systems of
Difference Equations 236
§4. Reducibility of Systems of Difference Equations on the Toroidal Set 245
§5. Reducibility of Nonlinear Systems of Difference Equations in the
Vicinity of the Toroidal Set 253
References 263
Index 279
PREFACE
The investigation of a great number of problems in celestial mechanics, physics and
engineering involves the study of oscillating systems governed by systems of nonlinear
ordinary differential equations or by systems of partial differential equations. The me-
thods of investigation of periodic and quasiperiodic solutions for these systems are high-
ly developed and presented in many fundamental monographs.
The recent growth of interest in the theory of differential equations with deviating ar-
gument and, especially, in the theory of differential equations with lag (and equations re-
ducible to these) has been caused by practical needs and, in the first place, by the rapid
development of the theory of automatic regulation systems. Equations with retarded ar-
gument have a wide spectrum of applications to the problems arising in automation, tele-
mechanics, radiolocation, electrical and radio communication, rocket engineering, ship-
building, theoretical cybernetics, medicine, biology, etc. (Zverkin, Kamensky, Norkin,
and Elsgoltz, 1962, 1963; Krasovsky, 1959; Martinyuk, 1971; Myshkis, 1972; Pelyukh
and Sharkovsky, 1974; Pinney, 1961).
Advances in technical sciences also contributed to the growth of interest in the dif-
ference equations which proved to be a convenient model for the description of pulse and
discrete dynamical systems and systems which include digital computing devices as a
component (Boltyansky, 1973; Halanay and Veksler, 1971). Moreover, difference equa-
tions are encountered when various classes of differential equations are solved numeri-
cally by the finite differences method.
The problems concerning the existence of periodic solutions and construction of al-
gorithms for calculating these solutions are among the most important aspects of the
investigation of nonlinear systems of differential equations with lag and various classes
of integro-differential and difference equations. Many well-known methods give good
solutions for weakly nonlinear systems with lag (Le., for systems with a small para-
meter). Among these one should mention Bogolyubov-Krylov's asymptotical method
(Bogolyubov and Mitropolsky, 1963; Volosov and Morgunov, 1971; Grebennikov, 1968;
Grebennikov and Ryabov, 1978, 1979; Krylov and Bogolyubov, 1937), Poincare's
method of a small parameter, Shimanov's method of auxiliary systems, and Lyapunov's
method of functional equations which was developed for weakly nonlinear systems by
Yu.A.Ryabov (see, e.g., (Bogolyubov, 1945; Bogolyubov, Mitropolsky, and Samoilenko,
1969; Martinyuk and Kozubovskaya, 1968; Martinyuk and Kolomiets, 1968; Martinyuk,
Mironov, and Kharabovskaya, 1971; Martinyuk and Fodchuk, 1963, 1966; Mitropolsky
and Martinyuk, 1969, 1979; Mitropolsky, 1964; Mitropolsky and Moseenkov, 1976;
Rubanik, 1969; Rozhkov, 1968, 1970; Ryabov, 1960a,b, 1961, 1962, 1964; Hale, 1966;
Shimanov, 1957, 1959, 1960, 1965, 1970».
xi