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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.
The theory of partial differential equations is a wide and rapidly
developing branch of contemporary mathematics. Problems related to
partial differential equations of order higher than one are so
diverse that a general theory can hardly be built up. There are
several essentially different kinds of differential equations
called elliptic, hyperbolic, and parabolic. Regarding the
construction of solutions of Cauchy, mixed and boundary value
problems, each kind of equation exhibits entirely different
properties. Cauchy problems for hyperbolic equations and systems
with variable coefficients have been studied in classical works of
Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic
equations were considered by Vishik, Ladyzhenskaya, and that for
general two dimensional equations were investigated by Bitsadze,
Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last
decade the theory of solvability on the whole of boundary value
problems for nonlinear differential equations has received
intensive development. Significant results for nonlinear elliptic
and parabolic equations of second order were obtained in works of
Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others.
Concerning the solvability in general of nonlinear hyperbolic
equations, which are connected to the theory of local and nonlocal
boundary value problems for hyperbolic equations, there are only
partial results obtained by Bronshtein, Pokhozhev, Nakhushev."
Many dynamical systems are described by differential equations that
can be separated into one part, containing linear terms with
constant coefficients, and a second part, relatively small compared
with the first, containing nonlinear terms. Such a system is said
to be weakly nonlinear. The small terms rendering the system
nonlinear are referred to as perturbations. A weakly nonlinear
system is called quasi-linear and is governed by quasi-linear
differential equations. We will be interested in systems that
reduce to harmonic oscillators in the absence of perturbations.
This book is devoted primarily to applied asymptotic methods in
nonlinear oscillations which are associated with the names of N. M.
Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. The advantages
of the present methods are their simplicity, especially for
computing higher approximations, and their applicability to a large
class of quasi-linear problems. In this book, we confine ourselves
basi cally to the scheme proposed by Krylov, Bogoliubov as stated
in the monographs 6,211. We use these methods, and also develop and
improve them for solving new problems and new classes of nonlinear
differential equations. Although these methods have many
applications in Mechanics, Physics and Technique, we will
illustrate them only with examples which clearly show their
strength and which are themselves of great interest. A certain
amount of more advanced material has also been included, making the
book suitable for a senior elective or a beginning graduate course
on nonlinear oscillations."
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.
Many dynamical systems are described by differential equations that
can be separated into one part, containing linear terms with
constant coefficients, and a second part, relatively small compared
with the first, containing nonlinear terms. Such a system is said
to be weakly nonlinear. The small terms rendering the system
nonlinear are referred to as perturbations. A weakly nonlinear
system is called quasi-linear and is governed by quasi-linear
differential equations. We will be interested in systems that
reduce to harmonic oscillators in the absence of perturbations.
This book is devoted primarily to applied asymptotic methods in
nonlinear oscillations which are associated with the names of N. M.
Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. The advantages
of the present methods are their simplicity, especially for
computing higher approximations, and their applicability to a large
class of quasi-linear problems. In this book, we confine ourselves
basi cally to the scheme proposed by Krylov, Bogoliubov as stated
in the monographs 6,211. We use these methods, and also develop and
improve them for solving new problems and new classes of nonlinear
differential equations. Although these methods have many
applications in Mechanics, Physics and Technique, we will
illustrate them only with examples which clearly show their
strength and which are themselves of great interest. A certain
amount of more advanced material has also been included, making the
book suitable for a senior elective or a beginning graduate course
on nonlinear oscillations."
The theory of partial differential equations is a wide and rapidly
developing branch of contemporary mathematics. Problems related to
partial differential equations of order higher than one are so
diverse that a general theory can hardly be built up. There are
several essentially different kinds of differential equations
called elliptic, hyperbolic, and parabolic. Regarding the
construction of solutions of Cauchy, mixed and boundary value
problems, each kind of equation exhibits entirely different
properties. Cauchy problems for hyperbolic equations and systems
with variable coefficients have been studied in classical works of
Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic
equations were considered by Vishik, Ladyzhenskaya, and that for
general two dimensional equations were investigated by Bitsadze,
Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last
decade the theory of solvability on the whole of boundary value
problems for nonlinear differential equations has received
intensive development. Significant results for nonlinear elliptic
and parabolic equations of second order were obtained in works of
Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others.
Concerning the solvability in general of nonlinear hyperbolic
equations, which are connected to the theory of local and nonlocal
boundary value problems for hyperbolic equations, there are only
partial results obtained by Bronshtein, Pokhozhev, Nakhushev."
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.
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