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Consider the planar differential systems where and are real polynomials in and the maximum degree of and is . The second half of the famous Hilbert’s 16th problem, proposed in 1900, can be stated as follows (see [70]):

In this chapter we will explain the relation between the number of zeros of the Abelian integrals and the number of limit cycles of the corresponding planar polynomial differential systems.

To study the weak Hilbert’s 16th problem by using Abelian integrals, it is crucial to estimate the number of zeros of the Abelian integral. In this chapter, we introduce several methods to study the number of zeros of the Abelian integral () given in (1.10), which is related to the codimension 2 Bogdanov-Takens bifurcation problem, as we explained in subsection 1.2.2.

As we explained in Subsection 1.2.1, any cubic generic Hamiltonian, with at least one period annulus contained in its level curves, can be transformed into the normal form where are parameters lying in the open region Figure 1 (in Subsection 1.2.1) shows all five possible phase portraits of in the generic cases. Here is the Hamiltonian vector field corresponding to , i.e., The vector field has a center at the origin in the ()-plane, and the continuous family of ovals, surrounding the center, is The oval shrinks to the center as → 0, and the oval terminates at the saddle loop of the saddle point (1, 0) when → 1/6.