STABILITY OF SINGULAR LIMIT CYCLES FOR ABEL EQUATIONS

摘要

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' = at(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.