Limit Cycles for a Class of Discontinuous Generalized Lienard Polynomial Differential Equations

We divide R2 in l sectors S1, ..., Sl, with l > 1 even. We define in R2 a discontinuous differential system such that in each sector Sk, for k = 1, ..., l, is defined a smooth generalized Lienard polynomial differential equation ẍ + fi(x)ẋ + gi(x) = 0, i = 1, 2 alternatively, where fi and gi are polynomials of degree n−1 and m respectively. We apply the averaging theory of first order for discontinuous differential systems to this class of non-smooth generalized Lienard polynomial differential systems and we show that for any n and m there are such non-smooth Lienard polynomial equations having at least max{n, m} limit cycles. Note that this number is independent of l. Roughly speaking this result shows that the non-smooth classical (m = 1) Lienard polynomial differential systems can have at least the double number of limit cycles than the smooth ones, and that the non-smooth generalized Lienard polynomial differential systems can have at least one more limit cycle than the smooth ones. Of course, these comparisons are done with the present known results.