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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Algebraic invariant curves for the Liénard equation
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by Henryk Żoła̧dek PDF
Trans. Amer. Math. Soc. 350 (1998), 1681-1701 Request permission

Abstract:

Odani has shown that if $\deg g\leq \deg f$ then after deleting some trivial cases the polynomial system $\dot {x}=y, \dot {y}=-f(x)y-g(x)$ does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of $\deg f$ and $\deg g$. We give also a simple presentation of Yablonsky’s example of a quartic limit cycle in a quadratic system.
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Additional Information
  • Henryk Żoła̧dek
  • Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
  • Email: zoladek@mimuw.edu.pl
  • Received by editor(s): April 10, 1995
  • Received by editor(s) in revised form: August 26, 1996
  • Additional Notes: Supported by Polish KBN Grant No 2 P03A 022 08
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 1681-1701
  • MSC (1991): Primary 34C05, 58F21
  • DOI: https://doi.org/10.1090/S0002-9947-98-02002-9
  • MathSciNet review: 1433130