Weak-foci of high order and cyclicity
Liang, Haihua (Guangdong Polytechnic Normal University(China). Department of Computer Science)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data:	2016
Resum:	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 QiuYan2009 where the highest weak-focus order is n^2 n-2 for n=4,6, 18. Moreover, we provide a system which has a weak-focus with order (n-1)^2 for n 12. We show that Christopher's approach Chr2006, 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.
Ajuts:	Ministerio de Economía y Competitividad MTM2008-03437 Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 316338 European Commission 316339
Nota:	Agraïments: This work was done when H. Liang was visiting the Department of Mathematics of Universitat Autònoma de Barcelona. He is very grateful for the support and hospitality. The first author is supported by the NSF of China (No. 11201086 and No. 11401255) and the Excellent Young Teachers Training Program for colleges and universities of Guangdong Province, China (No. Yq2013107).
Nota:	Agraïments: The second author is partially supported by UNAB13-4E-1604.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Cyclicity ; Lyapunov quantities ; Polynomial system ; Weak-focus order
Publicat a:	Qualitative theory of dynamical systems, 2016 , ISSN 1662-3592

DOI: 10.1007/s12346-016-0189-9

Postprint 12 p, 332.3 KB

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