Number of location of limit cycles in a class of perturbed polynomial systems (Q1780334)

By investigating the zeros of abelian integrals, the authors show that for \(\ell = 2n + 1\) and \(\ell = 2n + 2\), for \(0 < \epsilon \ll 1\) the perturbation \(x' = y + \epsilon \sum_{k = 0}^\ell a_k x^k\), \(y' = -x\) of the canonical linear center has at most \(n\) limit cycles. They study the cases \(n = 2\) and \(n = 3\) in detail, for example giving necessary and sufficient conditions for the appearance of three limit cycles when \(n = 3\).