\centerline{\bf Introduction}

\medskip

At the present time there exist two different approaches to the
investigation of dynamic systems. The first of them is the
classical theory
of the stability of motion. There are many books devoted to this
approach [1,5,\linebreak
7-9,12,18,29,30,42,47].
In the last decades the most in\-ten\-si\-ve work in this direction is
observed in the problems of me\-cha\-nics [17,31,40] and in the control
theory [4,14,17,25,\linebreak
31,37].

The second direction in the investigation of dynamic systems is
the theory of chaos, to which the books
[3,11,33,\linebreak
35,39,43,44,46,48,49] are devoted.

In the present work an attempt is made to combine various
elements of mathematical constructions, which had been
used in the framework of one of two approximate methods
mentioned.

It turned out to be that the definition of Lyapunov exponents via
singular values has a significant advantage in comparison with
the classical definition of Lyapunov exponents from the  point of
view of their estimation. This fact is demonstrated in
Chapters 1 and 2 for continuous and discrete systems.

In Chapter 3 by methods, which usually are used for the
estimation of dimensions of attractors, some new results are
obtained in the classical problem of stability by the first
approximation.

In Chapter 4 the methods, used previously to analyze the stability of
discrete control systems, are developed for obtaining frequency
con\-di\-ti\-ons of existence of strange attractors.

\newpage

This book is a natural continuation of the books [24,25]. In these books
the authors describe the frequency methods to be used here in
Chapters 1,2 and 4 in details. Therefore, the reader easily
becomes familiar with the methods which are set forth here,
if he is familiar with [24,25].

The main motivation to write this book was my opinion that all the
methods demonstated here may be developed and generalized in many
directions. Thus, the results ob\-ta\-i\-ned may be applied to
various systems having stable or chaotic character.

I am grateful to Vera B.Smirnova and Alexander I.She\-pe\-lja\-vyi for
their useful comment.
At first the mauscript was written in Russian and then has been
translated into English by Elmira A.Gurmuzova. The
camera-ready copy was prepared by Jury K.Zotov and Inga
I.Ryzhakova. I am thankful to them for this work.

The basic material of the book was presented in lectures which
were delivered by author during autumn semester in 1997 in the
Institute for Problems of Mechanical En\-ge\-ne\-e\-ring of the Russian
Academy of Sciences in the framework of the Programme of
Integration N 326.53 and N 589.
The lectures were delivered for students, post-grantuate
stu\-dents, and young scientists of various St.Petersburg
universities and institutes. This work was partially sup\-por\-ted by
grant of RFFI N 96-01-01172.



December 1997 \hfill G.A.Leonov

Faculty of Mathematics and Mechanics 

St.Petersburg State University

Bibliotechnaya pl. 2

198904 Petrodvorets, 
St.Petersburg, Russia

e-mail: leonov \@ math.lgu.spb.su

    


