Stability of singular limit cycles for Abel equations

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 Hopflike 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 familyx 2 , with a, b > 0, has at most two positive limit cycles for any t B , t A .

Rational limit cycles of Abel equations

“…where x ∈ [0, 1] and y are real variables and A(x) and B(x) are polynomials. The limit cycles of these equations have been intensively investigated mainly when the functions A(x) and B(x) are periodic (see for instance [1,2,3,4,5,6,9,11,14,15,17,18,19,20,21,23,24,25,26]), and also when A(x) and B(x) are polynomial (see for instance [10,12,13,16,22]). Here we are interested in the rational limit cycles of equation ( 1) when the functions A(x) and B(x) are polynomials.…”