**Translations of Mathematical Monographs**

1991;
288 pp;
Hardcover

MSC: Primary 34; 58;
Secondary 14; 41; 57

Print ISBN: 978-0-8218-4553-0

Product Code: MMONO/94

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

**Electronic ISBN: 978-1-4704-4506-5
Product Code: MMONO/94.E**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

# Finiteness Theorems for Limit Cycles

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*Yu S Il’yashenko*

This book is devoted to the following finiteness theorem: A polynomial vector field on the real plane has a finite number of limit cycles. To prove the theorem, it suffices to note that limit cycles cannot accumulate on a polycycle of an analytic vector field. This approach necessitates investigation of the monodromy transformation (also known as the Poincaré return mapping or the first return mapping) corresponding to this cycle. To carry out this investigation, this book utilizes five sources: The theory of Dulac, use of the complex domain, resolution of singularities, the geometric theory of normal forms, and superexact asymptotic series. In the introduction, the author presents results about this problem that were known up to the writing of the present book, with full proofs (except in the case of results in the local theory and theorems on resolution of singularities).

#### Reviews & Endorsements

Excellent book … devoted to a rigorous proof of this finiteness theorem, and some related results are proved along with it … this valuable and interesting book will give the readers a good understanding of this deep and elegant work, and … more and more mathematicians will be interested in solving Hilbert's difficult 16th problem.

-- Mathematical Reviews

The viewpoint is high and the techniques are delicate and profound.

-- Zentralblatt MATH

An indispensable component of a complete mathematical library … one is struck by the originality, the creative power, the depth of thought, and the technical facility demonstrated. In Professor Mauricio Peixoto's words, this is ‘mathematics of the highest order’. Many of those who love mathematics will find treasure here.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## Finiteness Theorems for Limit Cycles

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Introduction 112
- Chapter I. Decomposition of a monodromy transformation into terms with noncomparable rates of decrease 4152
- Chapter II. Function-theoretic properties of regular functional cochains 8192
- Chapter III. The Phragmén-Lindelöf theorem for regular functional cochains 121132
- Chapter IV. Superexact asymptotic series 159170
- Chapter V. Ordering of functional cochains on a complex domain 203214
- Bibliography 287298
- Back Cover Back Cover1301