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Introduction to the Qualitative Theory of Dynamical Systems on Surfaces
S. Kh. Aranson, Novgorod, Russia, G. R. Belitsky, Ben-Gurion University of the Negev, Beer Sheva, Israel, and E. V. Zhuzhoma, Nizhnii Novgorod University, Russia
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Translations of Mathematical Monographs
1996; 325 pp; hardcover
Volume: 153
ISBN-10: 0-8218-0369-7
ISBN-13: 978-0-8218-0369-1
List Price: US$129
Member Price: US$103.20
Order Code: MMONO/153
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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

"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

  • Dynamical systems on surfaces
  • Structure of limit sets
  • Topological structure of flows
  • Local structure of dynamical systems
  • Transformations of the circle
  • Classification of flows on surfaces
  • Relation between smoothness properties and topological properties of flows
  • Bibliography
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