The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign

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September 2009, 8(5): 1493-1501. doi: 10.3934/cpaa.2009.8.1493

The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign

Amelia Álvarez ^1, , José-Luis Bravo ^1, and Manuel Fernández ^1,

1.	Departamento de Matemáticas, Universidad de Extremadura, Badajoz, 06071, Spain, Spain, Spain

Received June 2008 Revised February 2009 Published April 2009

Abstract
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We study the number of limit cycles (isolated periodic solutions in the set of all periodic solutions) for the generalized Abel equation $x'=a(t)x^{n_a}+b(t)x^{n_b}+c(t)x^{n_c}+d(t)x$, where $n_a > n_b > n_c > 1$, $a(t),b(t),c(t), d(t)$ are $2\pi$-periodic continuous functions, and two of $a(t),b(t),c(t)$ have definite sign.
We obtain examples with at least seven limit cycles, and some sufficient conditions for the equation to have at most one or at most two positive limit cycles.

Keywords: limit cycle., Abel equation, periodic solution.

Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 37C27, 37G1.

Citation: Amelia Álvarez, José-Luis Bravo, Manuel Fernández. The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493

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