On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

Abstract

In this paper, we consider the bifurcation of limit cycles for generic L–V system ( ẋ=y+x2−y2±43xy,ẏ=−x+2xy) and B–T system (ẋ=y,ẏ=−x+x2) under perturbations of piecewise smooth polynomials with degree n. After linear transformation, we choose switching line y=0. By using Picard–Fuchs equations, we bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles (taking into account the multiplicity) for generic L–V system and B–T system are respectively 39n−72(n≥4),39,59,98(n=1,2,3) and 12n+3n2+5.