On the Distribution of Limit Cycles in a Lienard System with a Nilpotent Center and a Nilpotent Saddle

In this work, we study the Abelian integral I(h) corresponding to the following Lienard system, (x) over dot = y, (y) over dot = x(3)(x - 1)(3) + epsilon(a + bx + cx(3) + x(5)) y, where 0 < epsilon << 1, a, b and c are real bounded parameters. By using the expansion of I(h) and a new algebraic criterion developed in [ Grau et al., 2011], it will be shown that the sharp upper bound of the maximal number of isolated zeros of I(h) is 4. Hence, the above system can have at most four limit cycles bifurcating from the corresponding period annulus. Moreover, the configuration (distribution) of the limit cycles is also determined. The results obtained are new for this kind of Lienard system.