Limit cycles for some abel equations having coefficients without fixed signs

J. L. Bravo; M. FernÁndez; A. Gasull

doi:10.1142/S0218127409025195

Limit cycles for some abel equations having coefficients without fixed signs

J. L. Bravo, M. FernÁndez, A. Gasull

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

18 Citations (Scopus)

Abstract

We prove that some 2π-periodic generalized Abel equations of the form x′ = A(t)xn + B(t)xm + C(t)x, with n ≠ m and n, m < 2 have at most three limit cycles. The novelty of our result is that, in contrast with other results of the literature, our hypotheses allow the functions A,B, and C to change sign. Finally we study in more detail the Abel equation x′ = A(t)x3 + B(t)x2, where the functions A and B are trigonometric polynomials of degree one. © 2009 World Scientific Publishing Company.

Original language	English
Pages (from-to)	3869-3876
Journal	International Journal of Bifurcation and Chaos
Volume	19
Issue number	11
DOIs	https://doi.org/10.1142/S0218127409025195
Publication status	Published - 1 Jan 2009

Keywords

Abel equation
Limit cycle
Periodic solution

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title = "Limit cycles for some abel equations having coefficients without fixed signs",

abstract = "We prove that some 2π-periodic generalized Abel equations of the form x′ = A(t)xn + B(t)xm + C(t)x, with n ≠ m and n, m < 2 have at most three limit cycles. The novelty of our result is that, in contrast with other results of the literature, our hypotheses allow the functions A,B, and C to change sign. Finally we study in more detail the Abel equation x′ = A(t)x3 + B(t)x2, where the functions A and B are trigonometric polynomials of degree one. {\textcopyright} 2009 World Scientific Publishing Company.",

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T1 - Limit cycles for some abel equations having coefficients without fixed signs

AU - Bravo, J. L.

AU - FernÁndez, M.

AU - Gasull, A.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - We prove that some 2π-periodic generalized Abel equations of the form x′ = A(t)xn + B(t)xm + C(t)x, with n ≠ m and n, m < 2 have at most three limit cycles. The novelty of our result is that, in contrast with other results of the literature, our hypotheses allow the functions A,B, and C to change sign. Finally we study in more detail the Abel equation x′ = A(t)x3 + B(t)x2, where the functions A and B are trigonometric polynomials of degree one. © 2009 World Scientific Publishing Company.

AB - We prove that some 2π-periodic generalized Abel equations of the form x′ = A(t)xn + B(t)xm + C(t)x, with n ≠ m and n, m < 2 have at most three limit cycles. The novelty of our result is that, in contrast with other results of the literature, our hypotheses allow the functions A,B, and C to change sign. Finally we study in more detail the Abel equation x′ = A(t)x3 + B(t)x2, where the functions A and B are trigonometric polynomials of degree one. © 2009 World Scientific Publishing Company.

KW - Abel equation

KW - Limit cycle

KW - Periodic solution

U2 - 10.1142/S0218127409025195

DO - 10.1142/S0218127409025195

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SP - 3869

EP - 3876

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ER -