Stability of singular limit cycles for Abel equations

José Luis Bravo,Manuel Fernández,Armengol Gasull

doi:10.3934/dcds.2015.35.1873

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 $t_B,t_A$.

Highlights

Introduction and Main resultsThe study of the number of periodic solutions of Abel differential equations is a challenging question
These equations are interesting because they provide models of real phenomena, see for instance [4, 9, 12], or as a tool for studying several subcases of Hilbert XVI problem on the number of limit cycles of planar polynomial differential equations, see [7, 14]
In this paper we consider simple Abel equations for which there is no uniqueness of positive limit cycles and study their number by controlling the nature of the double closed solutions

Summary

Introduction and Main results

The study of the number of periodic solutions of Abel differential equations is a challenging question. In this paper we consider simple Abel equations for which there is no uniqueness of positive limit cycles and study their number by controlling the nature of the double closed solutions. In case (2), it is proved in [1] that if for some α, β ∈ R the function αA + βB does not vanish identically and does not change sign in (0, 1) the Abel equation has at most one positive limit cycle. Notice that the above definition of monotonic with respect to λ for families of Abel equations is an adaptation to this setting of the so called rotated families of planar vector fields introduced by Duff in 1953, see [8] or [18, Sec. 4.6] For these families of vector fields, the control of semistable bifurcations of limit cycles is crucial for understanding their global bifurcation diagram of limit cycles

Proofs of the main results

Example of application