Stability of singular limit cycles for Abel equations

© 2015 Discrete and Continuous Dynamical Systems. We obtain a criterion for determining the stability of singular limit cycles of Abel equations x′ = A(t)x3 + B(t)x2. 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′ = at(t - tA)x3 + b(t - tB)x2, with a, b > 0, has at most two positive limit cycles for any tB, tA.