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| 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 $t_A,t_B$. |
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