Weak-foci of High Order and Cyclicity

A particular version of the 16th Hilbert’s problem is to estimate the number, M(n), of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for M(n) for some concrete n by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high cyclicity, we search for systems with the highest possible weak-focus order. For even n, the studied polynomial system of degree n was the one obtained by [20] where the highest weak-focus order is n + n − 2 for n = 4, 6, . . . , 18. Moreover, we provide a system which has a weak-focus with order (n− 1) for n ≤ 12. We show that Christopher’s approach [5], aiming to study the cyclicity of centers, can be applied also to the weak-focus case. We also show by concrete examples that, in some families, this approach is so powerful and the cyclicity can be obtained in a simple computational way. Finally, using this approach, we obtain that M(6) ≥ 39,M(7) ≥ 34 and M(8) ≥ 63.