Limit cycles for discontinuous generalized lienard polynomial differential equations

We divide ℝ2into sectors S1,...,Sl, with l > 1 even, and define a discontinuous differential system such that in each sector, we have a smooth generalized Lienard polynomial differential equation ẍ + fi(x)x ̇ + gi(x) = 0, i = 1,2 alternatively, where fiand giare polynomials of degree n - 1 and m respectively. Then we apply the averaging theory for first-order discontinuous differential systems to show that for any n and m there are non-smooth Lienard polynomial equations having at least max{n, m} limit cycles. Note that this number is independent of the number of sectors. 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.2000 Mathematics Subject Classification. © 2013 Texas State University - San Marcos.