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Journal of Dynamical Systems and Geometric Theories

(Since 2002)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

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Journal of Dynamical Systems and Geometric Theories
Open Access Research Article

Rational Periodic Solutions on Some Generalized Abel Equations

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pp. 177–189Vol. 20Issue 2July 2022DOI: 10.1080/1726037X.2022.2142353XML
Received:
19 Jun 2022
Accepted:
10 Aug 2022
Published Online:
13 Nov 2022
Article type:
Research Article
Language:
EN
Article no.:
1726037X.2022.2142353
Pages:
177–189

Abstract

In this paper we deal with the equations a(x)dy/dx = A(x)y2 + B(x)y3, where a(x), A(x) and B(x) are complex polynomials with a(x)B(x) ≢ 0 and a(x) non-constant. First we show that the unique rational limit cycles that these equations can have are of the form y = 1/p(x) being p(x) some polynomial. Second we provide an upper bound on the number of these rational limit cycles. Moreover, we prove that if deg(B(x)) − deg(a(x)) + 1 is odd, or deg(A) > (deg(B(x)) + deg(a(x)) − 1)/2, then these Abel equations have at most two rational limit cycles and we provide examples of these Abel equations with three nontrivial rational periodic solutions.

Keywords

Algebraic periodic solutionsRational periodic solutionsAbel equationsgeneralized Abel equations

Subject Classifications

34C0534C0734C08

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