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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer,*Qualitative theory of second-order dynamic systems*, Halsted Press (A division of John Wiley & Sons), New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian by D. Louvish. MR**0350126****2.**V. Arnol′d,*Chapitres supplémentaires de la théorie des équations différentielles ordinaires*, “Mir”, Moscow, 1980 (French). Translated from the Russian by Djilali Embarek. MR**626685****3.**Marc Carbonell and Jaume Llibre,*Limit cycles of polynomial systems with homogeneous nonlinearities*, J. Math. Anal. Appl.**142**(1989), no. 2, 573–590. With the collaboration of Bartomeu Coll. MR**1014597**, 10.1016/0022-247X(89)90021-8**4.**Chong Qing Cheng and Yi Sui Sun,*Metamorphoses of phase portraits of vector field in the case of symmetry of order 4*, J. Differential Equations**95**(1992), no. 1, 130–139. MR**1142279**, 10.1016/0022-0396(92)90045-O**5.**CHERKAS, L.A. Number of limit cycles of an autonomous second-order system. Diff. Equations,**5**(1976), 666-668.**6.**Shui-Nee Chow, Cheng Zhi Li, and Duo Wang,*Normal forms and bifurcation of planar vector fields*, Cambridge University Press, Cambridge, 1994. MR**1290117****7.**B. Coll, A. Gasull, and R. Prohens,*Differential equations defined by the sum of two quasi-homogeneous vector fields*, Canad. J. Math.**49**(1997), no. 2, 212–231. MR**1447489**, 10.4153/CJM-1997-011-0**8.**A. Gasull and J. Llibre,*Limit cycles for a class of Abel equations*, SIAM J. Math. Anal.**21**(1990), no. 5, 1235–1244. MR**1062402**, 10.1137/0521068**9.**John Guckenheimer,*Phase portraits of planar vector fields: computer proofs*, Experiment. Math.**4**(1995), no. 2, 153–165. MR**1377416****10.**E. I. Horozov,*Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3*, Trudy Sem. Petrovsk.**5**(1979), 163–192 (Russian). MR**549627****11.**Bernd Krauskopf,*Bifurcation sequences at 1:4 resonance: an inventory*, Nonlinearity**7**(1994), no. 3, 1073–1091. MR**1275542****12.**N. G. Lloyd,*A note on the number of limit cycles in certain two-dimensional systems*, J. London Math. Soc. (2)**20**(1979), no. 2, 277–286. MR**551455**, 10.1112/jlms/s2-20.2.277**13.**A. I. Neĭshtadt,*Bifurcations of the phase pattern of an equation system arising in the problem of stability loss of self-oscillations close to 1:4 resonance*; Russian transl., J. Appl. Math. Mech.**42**(1978), no. 5, 896–907 (1979). MR**620880****14.**P. Yu, M. Han, and Y. Yuan,*Analysis on limit cycles of 𝑍_{𝑞}-equivariant polynomial vector fields with degree 3 or 4*, J. Math. Anal. Appl.**322**(2006), no. 1, 51–65. MR**2238147**, 10.1016/j.jmaa.2005.08.068**15.**André Zegeling,*Equivariant unfoldings in the case of symmetry of order 4*, Serdica**19**(1993), no. 1, 71–79. MR**1241150**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
34C07,
34C14,
34C23,
37C27

Retrieve articles in all journals with MSC (2000): 34C07, 34C14, 34C23, 37C27

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

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.