Limit cycles in discontinuous classical Liénard equations

We study the number of limit cycles which can bifurcate from the periodic orbits of a linear center perturbed by nonlinear functions inside the class of all classical polynomial Liénard differential equations allowing discontinuities.In particular our results show that for any n≥1n≥1 there are differential equations of the form ẍ+f(x)ẋ+x+sgn(ẋ)g(x)=0, with ff and gg polynomials of degree nn and 11 respectively, having [n/2]+1[n/2]+1 limit cycles, where [⋅][⋅] denotes the integer part function.