On the number of limit cycles of a class of polynomial systems of Liénard type

In this paper we study a planar system of the form x ̇ = p k ( y ) , y ̇ = − g m ( x ) − ε f n ( x ) y , where p k ( y ) is a polynomial of degree k in y , and f n ( x ) , g m ( x ) are polynomials in x with degree of n and m , respectively. Let H ( m , n , k ) denote the maximal number of limit cycles having odd multiplicities of this system. We give some lower bounds of H ( m , n , k ) .