Limit cycles of non-autonomous scalar ODEs with two summands

José-Luis Bravo; Manuel Fernández

doi:10.3934/cpaa.2013.12.1091

Article Contents

2013, Volume 12, Issue 2: 1091-1102. Doi: 10.3934/cpaa.2013.12.1091

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Limit cycles of non-autonomous scalar ODEs with two summands

José-Luis Bravo^1, and
Manuel Fernández^2,

1.
Departamento de Matemáticas, Universidad de Extremadura, Badajoz, 06071
2.
Departamento de Matemáticas, Universidad de Extremadura, Facultad de Ciencias, 06071 Badajoz

Received: July 31, 2011

Revised: December 31, 2011

Published: February 2013

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Abstract

We establish upper bounds for the number of limit cycles (isolated periodic solutions in the set of periodic solutions) of the two families of scalar ordinary differential equations $x'=(a(t) x +b(t)) f(x)$ and $x'=a(t) g(x) +b(t)f(x)$, where $f(x)$ and $g(x)$ are analytic funtions and $a(t)$, $b(t)$ are $T$--periodic continuous functions for which there exist $\alpha, \beta \in R$ such that $\alpha a(t)+\beta b(t)$ is not identically zero and does not change sign in $[0,T]$. As a consequence we obtain that generalized Abel equations $x'=a(t)x^n + b(t)x^m$, where $n> m \geq 1$ are natural numbers, have at most three limit cycles.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34C07.

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