INTRODUCTION

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Among processes studied by various natural sciences (mechanics, physics,

ecology, etc.), an important place is occupied by oscillation processes. By now,

numerous efficient methods for the investigation of oscillation phenomena described

by linear and nonlinear differential equations have been developed and

mathematically justified [AVK, Bib, BNF, KBK, LiR, MiR, Nei, Pero, Pli1, Pros,

FiS, Hal2]. It turned out that, among these methods, the most efficient are asymptotic

methods, in particular, the averaging method, the method of integral manifolds,

and iterative methods developed by mathematicians of the Kiev Mathematical

School (Krylov, Bogolyubov, Mitropol’skii, Samoilenko, and their disciples)

[Bog, BoZ, BoM1, BoM2, BMS, GGP1, GGP2, GoP, Gre, GrR1–GrR3, KrB,

Kul, Luc, Lyk, LyB, MaS, MaT, Mit1–Mit4, MiLo, MiLy, MiS1–MiS5, MSK,

MSM1, MSM2, MiK, Par, Pere, PeA, Pet1–Pet10, PeL, PeP, Sam1–Sam10, SaP1,

SaP2, SPe1–SPe7, SPet1, SPet2, SaR, SaS, SaT, SSh, TeA, Tro, FBKCY, SVL,

SSY].

The foundations of the averaging method were laid in works of the founders

of celestial mechanics in the times of Lagrange and Laplace. The idea of this

method is as follows: Using a special operator, one replaces the system of differential

equations under investigation by another system (the so-called averaged

system). The averaged system, on the one hand, should be simpler in a certain

sense than the original one, and, on the other hand, it must describe the main features

of the phenomenon under investigation. In this case, there naturally arises

the problem of justification of the averaging method, i.e., the problem of finding

efficient estimates for the norm of the difference of solutions of the original and

the averaged equations on a finite or an infinite time interval.

Although the averaging method has been used for the solution of numerous

problems for almost two centuries, the problem of its justification remained unsolved

for a long time. Only in the 1930–1940s, beginning with Fatou’s work

[Fat], the first fundamental results were obtained in this direction. Thus, Bogolyubov

showed [Bog, Mit2] that, for systems of the standard form, the averaging

1

2 Introduction

method is closely related to the problem of the existence of a change of variables

that enables one to exclude the time variable on the right-hand side of the

system. Furthermore, Bogolyubov investigated systems of equations of higher approximations

whose solutions approximate the solutions of the original system of

equations to within values proportional to integer powers of a small parameter ε.

The averaging method was further developed by Mitropol’skii and other authors

for various classes of differential equations with large and small parameters.

In particular, it was extended to equations with nondifferentiable right-hand sides,

integro-differential and stochastic differential equations, partial differential equations,

and linear differential equations with slowly varying parameters that describe

nonstationary oscillation processes. These results are presented in [Mit2].

Mitropol’skii and Samoilenko developed the axiomatic theory of the averaging

method [MiS4], which includes the classical version of the averaging method and,

in particular, leads to the method of normal forms [Bry1]. Important results on the

problem of justification of the averaging method were also obtained in [Aku, Vol,

VoM, ZaL, MiK, Plo, PlB, PlZ, PlL, Sam1, Sam5, Fil, Kha1, Kha2, KhF].

The last decades were marked by the extensive investigation of multifrequency

nonlinear systems of differential equations appearing in various problems of classical

and celestial mechanics, radio engineering, and physics. In this connection,

the development of algorithms for the asymptotic integration of oscillation systems

with many degrees of freedom and their mathematical justification has become

an urgent problem.

In the case of systems with constant frequency vector, this important problem

was solved by Mitropol’skii and Samoilenko in [MiS1–MiS5, Sam2–Sam4]. In

particular, they thoroughly investigated an important phenomenon appearing in

multifrequency systems, namely, quasiperiodic oscillations.

For systems with variable frequency vector, the best-studied cases are the oneand

two-dimensional cases, which were investigated by Arnol’d [Arn2, Arn4],

Bakhtin [Bak2], Neishtadt [Neis1], and Pronchatov [Pron]. In the works indicated,

efficient estimates were obtained for the error of the averaging method on a

finite time segment and for the measure of the set of initial data for the equations

under investigation for which the resonance phenomenon takes place. If the number

of frequencies is greater than two, then the investigation of oscillation systems

leads to considerable difficulties because, in this case, the structure of resonance

surfaces is very complicated [Arn4]. Certain problems of justification of averaging

schemes in multifrequency systems and their applications to the solution

of practical problems were studied by Anosov [3], Bakhtin [Bak1], Grebenikov

[Gre], Neishtadt [Neis2], Plotnikov [Plo, PlL], and Khapaev [Kha1, Kha2].

Introduction 3

The problem of averaging in Hamiltonian systems has been fairly thoroughly

studied. Note that, for such systems, a solution of the averaged equations for slow

variables is always stationary. Kolmogorov [Kol] and Arnol’d [Arn1] solved the

problem of the stability of Hamiltonian systems on an infinite time segment, and

Nekhoroshev [Nek1, Nek2] obtained an exponential estimate for stability time for

almost all Hamiltonian functions.

The averaging method is often used for the solution of boundary-value problems

appearing in the simulation of the behavior of real processes and in problems

of optimal control. In the course of investigation of such processes, in many cases

it is rather difficult to specify initial data that uniquely describe the process (the

Cauchy problem), but it is possible to determine the values of some parameters of

this process at certain times by using various devices. There is a fairly rich theory

of multipoint boundary-value problems for ordinary differential equations developed

in works of many authors [Boi, VaK, VaB, VaD, Gab, DmK, ZhK, Kig, LeL,

Luc, Pta, SaR, ChH]. This theory is based on both analytic methods, by using

which one studies the problem of the existence and uniqueness of solutions and

their continuous dependence on parameters, and numerical methods, which enable

one to calculate approximate values of solutions. Analytic and numerical methods

are often combined, which allows one to efficiently solve the problem of the

existence of solutions and their construction. An extensive bibliography on this

problem can be found in [SaR]. Applications of the averaging method to the solution

of boundary-value problems were studied by Akulenko [Aku], Chernous’ko

[Che], Bainov and Milusheva [BaM], Plotnikov, Zverkova, and Bardai [Plo, PlB,

PlZ], and others.

Another powerful and convenient method for the investigation of nonlinear

systems of differential equations is the method of integral manifolds. The first

deep results on integral manifolds of toroidal type were obtained by Krylov and

Bogolyubov [Bog, BoM1, KrB] in the process of justification of asymptotic methods

in nonlinear mechanics. Later, the ideas of these works were generalized in

[BoM2] and extensively developed in the study of differential equations in various

functional spaces. They also affected the character of new developments in perturbation

theory for toroidal manifolds and led to deep results of Diliberto [Dil1,

Dil2], Hale [Hal1], Moser [Mos1–Mos5], Sacker [Sac1, Sac2], and Sell [Sel1,

Sel2].

A new impulse toward the development of the theory of perturbations and

stability of invariant manifolds was given by the concept of Green function in the

problem of invariant tori of a linear extension of a dynamical system on a torus,

which led to new results in this theory [Sam8].

4 Introduction

The method of integral manifolds was extended to systems of differential

equations with slow and fast variables (in particular, singularly perturbed ones),

impulsive systems, systems close to integrable ones, etc. Important results concerning

the existence and properties of integral manifolds can be found in the

monographs of Bakai and Stepanovskii [BaS], Bibikov [Bib], Mitropol’skii and

Lykova [MiLy], Pliss [Pli2], Samoilenko and Perestyuk [SaP2], and Strygin and

Sobolev [StS].

In the present work, we investigate multifrequency nonlinear systems of ordinary

differential equations of the form

dx

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(x, τ )

ε

+ b (x, ϕ, τ, ε),

(1)

where x and ϕ are n- and m-dimensional vectors, respectively, τ is “slow”

time, ε is a small positive parameter, and the real vector functions a, b, and

ω belong to certain classes of smooth functions 2π-periodic in ϕ. Systems of

this type appear in the course of investigation of oscillation processes in numerous

problems of mechanics, electrical engineering, biology, etc. [GrR2, GrR3,

Mit3, Mit4, Hal2]. The main problem arising in the study of properties of solutions

of system (1) is the problem of resonance relations between the components

of the variable frequency vector ω(x, τ ). Here, the resonance case is understood

as the case where the scalar product of the vector ω(x, τ ) and a nonzero vector

with integer-valued coordinates turns into zero or becomes close to zero for

certain values of x and τ. This leads to the appearance of slowly varying harmonics

in the Fourier series on the right-hand sides of Eqs. (1) and generates the

problem of small denominators [Arn1, BMS, GrR2]. At present, there is a fairly

complete and rich theory of one- and two-frequency systems. Note that, in the

case of two-frequency systems, the resonance surfaces form, generally speaking,

a collection of level surfaces; therefore, the main effect in these systems is the

passage through resonances in the course of evolution. If an oscillation system

has a greater number of frequencies, then its solutions can stay in a neighborhood

of resonance surfaces for a fairly long time or intersect these surfaces at arbitrarily

small angles, which substantially complicates the investigation of oscillations.

The case m ≥ 3 has been studied to a significantly lesser extent than the oneand

two-frequency cases, and, therefore, the investigation of various aspects of the

theory of multifrequency nonlinear oscillation systems is an urgent problem. The

present monograph is devoted to the solution of certain problems in this theory.

Introduction 5

In Chapter 1, we establish uniform estimates for certain oscillation integrals

depending on parameters, which are used in the proof of new theorems on the

justification of the averaging method. The averaged (with respect to all angular

variables ϕ) system corresponding to system (1) has the form

dx

dτ

= a (x, τ, ε),

dϕ

dτ

= ω(x, τ )

ε

+ b (x, τ, ε),

(2)

where

_

a; b

_

= (2π)−m

_2π

0

. . .

_2π

0

_

a(x, ϕ, τ, ε); b(x, ϕ, τ, ε)

_

dϕ1 . . . dϕm.

The averaged system (2) is simpler than (1) because it does not contain oscillation

terms on its right-hand side, and, therefore, for the construction of its solution, one

can use numerical methods with step greater than in the case of (1). The problem

of the justification of the averaging method reduces to the proof of the estimate

_x(τ, ε) − x(τ, ε)_ + _ϕ(τ, ε) − ϕ(τ, ε)_ ≤ Cεα

or

_x(τ, ε) − x(τ, ε)_ ≤ Cεα

for all τ ∈ I. Here, C and α are certain positive constants, x(0, ε) = x(0, ε),

ϕ(0, ε) = ϕ(0, ε), and either I = [0, L], or I = [0, T(ε)] ( T(ε) → ∞ as

ε → 0), or I = [0,∞). We obtain efficient estimates for partial derivatives of the

difference of solutions of systems (1) and (2) with respect to the initial data, prove

an analog of Banfi–Filatov theorem [Fil, Ban], and investigate multifrequency

systems of higher approximations.

Chapter 2 is devoted to the application of the averaging method to the solution

of boundary-value problems. In the case of oscillation systems with ω = ω(τ ),

we prove the solvability of two-point boundary-value problems and establish the

quantitative dependence of the norm of the difference of solutions of original and

averaged problems on the value of the small parameter ε. Combining the averaging

method with the solution of boundary-value problems, we prove the existence

of solutions of system (1) defined on the entire axis the slow variables x(τ, ε) of

which are uniformly bounded. It is important to note that this result is established

6 Introduction

without using the method of integral manifolds, which requires additional restrictions

on a multifrequency system. The solvability of multipoint boundary-value

problems in the case ω = ω(x, τ ) is studied, and the existence of solutions of

boundary-value problems with parameters is proved.

In Chapter 3, we establish conditions for the existence of an integral manifold

of system (1) in the case where the frequencies of the system depend on τ. The

smoothness properties are studied, estimates for partial derivatives of the function

defining the integral manifold are obtained, and a theorem on the conditional

asymptotic stability of an integral manifold is proved. In a small neighborhood of

an asymptotically stable integral manifold x = X (ψ, τ, ε), we decompose the

equations for slow and fast variables, i.e., we construct a change of variables

x = y + X (ϕ, τ, ε), ϕ= ψ +Φ(y, ψ, τ, ε)

that reduces system (1) to the form

dy

dτ

= Y (y, ψ, τ, ε),

dψ

dτ

= ω(τ )

ε

+ b

_

X(ψ, τ, ε), ψ, τ, ε

_

.

The results obtained are used for the investigation of a system of weakly connected

oscillators with slowly varying frequencies.

In Chapter 4, we study the behavior of a dynamical system

dx

dt

= X(x), x∈ Rn, (3)

in a neighborhood of a toroidal manifold M filled by a quasiperiodic trajectory

of the system. By passing to local coordinates, we establish conditions for the

reducibility of system (3) in such a neighborhood to a system with quasiperiodic

coefficients and investigate the smoothness of the corresponding change of variables.

We prove a statement on the exponential attraction as t→∞ of a solution

of system (3) originating in a small neighborhood of the invariant manifold to the

corresponding solution of this system that lies on M. We also establish the invariance

of the behavior of trajectories of a dynamical system in a neighborhood

of the manifold M under small perturbations of system (3). The results obtained

are extended to the case of discrete dynamical systems.

Introduction 7

The present monograph is based on the investigations carried out by the authors

themselves [Pet1–Pet10, Sam5–Sam7, Sam9, SPe1–SPe7] and in collaboration

with their disciples [PeL, PeP, SPet1, SPet2].

The authors hope that the ideas and methods proposed in this monograph will

be further developed and applied to new classes of problems in the theory of nonlinear

oscillations.

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