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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points


Authors: M. J. Álvarez, A. Gasull and R. Prohens
Journal: Proc. Amer. Math. Soc. 136 (2008), 1035-1043
MSC (2000): Primary 34C07, 34C14; Secondary 34C23, 37C27
Published electronically: November 30, 2007
MathSciNet review: 2361879
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Abstract: In this paper we study those cubic systems which are invariant under a rotation of $ 2\pi/4$ radians. They are written as $ \dot{z}=\varepsilon z+p\,z^2\bar{z}-\bar{z}^3,$ where $ z$ is complex, the time is real, and $ \varepsilon=\varepsilon_1+i\varepsilon_2$, $ p=p_1+ip_2$ are complex parameters. When they have some critical points at infinity, i.e. $ \vert p_2\vert\le 1$, it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. $ \vert p_2\vert>1,$ there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.


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Additional Information

M. J. Álvarez
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain
Email: chus.alvarez@uib.es

A. Gasull
Affiliation: Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: gasull@mat.uab.cat

R. Prohens
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain
Email: rafel.prohens@uib.cat

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09072-7
PII: S 0002-9939(07)09072-7
Keywords: Planar autonomous ordinary differential equations, symmetric cubic systems, limit cycles
Received by editor(s): March 24, 2006
Received by editor(s) in revised form: January 16, 2007
Published electronically: November 30, 2007
Additional Notes: The first two authors were partially supported by grants MTM2005-06098-C02-1 and 2005SGR-00550. The third author was supported by grant UIB-2006. This paper was also supported by the CRM Research Program: On Hilbert’s 16th Problem.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.