Weak-Foci of High Order and Cyclicity

© 2016, Springer International Publishing. 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 Qiu and Yang (J Differ Equ 246:3361–3379, 2009) where the highest weak-focus order is n2+ n- 2 for n= 4 , 6 , … , 18. Moreover, we provide a system which has a weak-focus with order (n- 1) 2 for n≤ 100. We show that Christopher’s approach (Differ Equ Symb Comput Trends Math 30:23–35, 2006), 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.