Stability of singular limit cycles for Abel equations

José Luis Bravo; Manuel Fernández; Armengol Gasull

doi:10.3934/dcds.2015.35.1873

Article Contents

2015, Volume 35, Issue 5: 1873-1890. Doi: 10.3934/dcds.2015.35.1873

This issue Previous Article A variational approach to reaction-diffusion equations with forced speed in dimension 1 Next Article Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

Stability of singular limit cycles for Abel equations

1.
Departamento de Matemáticas, Universidad de Extremadura, Badajoz 06006
2.
Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received: March 2014

Revised: September 2014

Published: May 2017

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Abstract

We obtain a criterion for determining the stability of singular limit cycles of Abel equations $x'=A(t)x^3+B(t)x^2$. This stability controls the possible saddle-node bifurcations of limit cycles. Therefore, studying the Hopf-like bifurcations at $x=0$, together with the bifurcations at infinity of a suitable compactification of the equations, we obtain upper bounds of their number of limit cycles. As an illustration of this approach, we prove that the family $x'=a t(t-t_A)x^3+b (t-t_B)x^2$, with $a ,b>0$, has at most two positive limit cycles for any $t_B,t_A$.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 37C27, 37G15.

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References

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