**Translations of Mathematical Monographs**

1996;
325 pp;
Hardcover

MSC: Primary 58;
Secondary 34; 57; 54

Print ISBN: 978-0-8218-0369-1

Product Code: MMONO/153

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

**Electronic ISBN: 978-1-4704-4568-3
Product Code: MMONO/153.E**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

# Introduction to the Qualitative Theory of Dynamical Systems on Surfaces

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*Samuel Kh. Aranson; G. R. Belitsky; E. V. Zhuzhoma*

This book is an introduction to the qualitative theory
of dynamical systems on manifolds of low dimension (on the circle and
on surfaces). Along with classical results, it reflects the
most significant achievements in this area obtained in recent times
by Russian and foreign mathematicians whose work has not yet appeared
in the monographic literature. The main stress here is put on
global problems in the qualitative theory of flows on surfaces.

Despite the fact that flows on surfaces have the same local structure
as flows on the plane, they have many global properties intrinsic to
multidimensional systems. This is connected mainly with the existence of
nontrivial recurrent trajectories for such flows. The investigation of
dynamical systems on surfaces is therefore a natural stage in the
transition to multidimensional dynamical systems.

The reader of this book need be familiar only with basic courses
in differential equations and smooth manifolds. All the main
definitions and concepts required for understanding the contents are
given in the text.

The results expounded can be used for investigating
mathematical models of mechanical, physical, and other systems
(billiards in polygons, the dynamics of a spinning top with
nonholonomic constraints, the structure of liquid crystals, etc.).

In our opinion the book should be useful not only to
mathematicians in all areas, but also to specialists with a mathematical
background who are studying dynamical processes: mechanical
engineers, physicists, biologists, and so on.

#### Readership

Graduate students and researchers working in dynamical systems and differential equations, as well as specialists with a mathematical background who are studying dynamical processes: mechanical engineers, physicists, biologists, etc.

#### Reviews & Endorsements

These and many other wonders are revealed in this thorough monograph. Lovers of dynamical systems will find this a mine of interesting information.

-- Bulletin of the London Mathematical Society

Consists of seven well-written chapters with mathematical rigor, and only prerequisite knowledge of topology and differential equations on the level of undergraduate students is assumed … contains … not only rich material for studying dynamical systems of two-dimensional manifolds, but also a natural background for understanding properties of multidimensional dynamical systems.

-- Zentralblatt MATH

Comprehensive … serves as a good reference for flows on surfaces, and would be well suited for a specialized graduate course on these topics … very well written.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to the Qualitative Theory of Dynamical Systems on Surfaces

- Cover Cover11
- Title page v6
- Contents vii8
- Foreword xiii14
- Chapter 1. Dynamical systems on surfaces 116
- Chapter 2. Structure of limit sets 4560
- Chapter 3. Topological structure of a flow 101116
- Chapter 4. Local structure of dynamical systems 147162
- Chapter 5. Transformations of the circle 189204
- Chapter 6. Classification of flows on surfaces 239254
- Chapter 7. Relation between smoothness properties and topological properties of flows 309324
- Bibliography 321336
- Back Cover Back Cover1343