Weak-Foci of High Order and Cyclicity
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Haihua Liang [1] ; Joan Torregrosa [2]
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[1] Guangdong Polytechnic Normal University
Guangdong Polytechnic Normal University
China
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[2] Universitat Autònoma de Barcelona
Universitat Autònoma de Barcelona
Barcelona, España
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- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 16, Nº 2, 2017, págs. 235-248
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
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.
