ABSTRACT
The second part of Hilbert 16th problem is: “find the maximal number and relative positions of the limit cycles of a polynomial vector field of degree n”. The answer to this question is still open even in the case n = 2. Moreover let H(x,y) be real polynomial of degree 3, there are altogether 11 classes of quadratic Hamiltonian vector field XH , which possess the compact component of H −1(h) topologically (cf. [1]). It is still an open question that the lowest upper bound of the number of limit cycles of XH is under quadratic nonconservative perturbation. The question was extensively studied in many papers ([2], [3], [7]). In the paper we concentrate our efforts on a class of nongeneric Hamiltonian system with center and invariant straight line which connects two saddles. We consider a 4-parameter quadratic nonconservative unfolding of that, and get the possible bifurcation phenomenon of the nongeneric Hamiltonian system under any quadratic small perturbations in finite plane when the first Melnikov function is not zero.
