Limit cycle bifurcations by perturbing a class of planar quintic vector fields (Q2003948)

In the qualitative theory of differential systems the main problems are to obtain the phase portraits, the singular points and to determine the maximum number of limits cycles that these systems can have. The problem of establishing an upper bound for the maximum number of limits cycles has been the subject of many papers but it is still open. Due to the difficulty of this problem several researchers study the lower or upper bounds for the maximum number of limit cycles of particular differential systems. In this paper, the authors consider a concrete class of planar quintic vector fields and determine all the phase portraits. Moreover, they perturbe this vector field with polynomials of degree \(n\), \(n\in{\mathbb{N}}\) up to first order in \(\epsilon\). Using the corresponding abelian integrals they prove that \(3\left[ \tfrac{n-1}{2}\right]\), \(n\geq5\), is a lower bound for the maximum number of limit cycles bifurcating from the periodic orbits.