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Finiteness Theorems for Limit Cycles
Yu. S. Il'yashenko

Translations of Mathematical Monographs
1991; 288 pp; hardcover
Volume: 94
ISBN-10: 0-8218-4553-5
ISBN-13: 978-0-8218-4553-0
List Price: US$120
Member Price: US$96
Order Code: MMONO/94
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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).


"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

  • Decomposition of a monodromy transformation into terms with noncomparable rates of decrease
  • Function-theoretic properties of regular functional cochains
  • The Phragmén-Lindelöf theorem for regular functional cochains
  • Superexact asymptotic series
  • Ordering of functional cochains on a complex domain
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