LIMIT CYCLES BIFURCATED FROM A KIND OF PIECEWISE SMOOTH GENERALIZED ABEL EQUATION VIA THE FIRST ORDER MELNIKOV ANALYSIS

The study of the existence and distribution of limit cycles for generalized Abel equations comes from the famous Small-Pugh problem, which has been extended to non-smooth case. In this paper, we consider a kind of piecewise smooth generalized Abel equation with the separation line $ t=0 $. We are interested in its number of nontrivial limit cycles which are bifurcated from the periodic annulus of unperturbed equation. Under the first order Melnikov analysis, we show that the upper bound of this kind nontrivial limit cycles is $ 2(m+1) $ if $ p $ is odd, and $ m+1 $ if $ p $ is even. The upper bound in both cases can be reached separately.