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

DOI:
https://doi.org/10.1090/S0002-9939-07-09072-7

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 radians. They are written as where is complex, the time is real, and , are complex parameters. When they have some critical points at infinity, i.e. , 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. 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:
https://doi.org/10.1090/S0002-9939-07-09072-7

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.