##
**The center-focus problem and reversibility.**
*(English)*
Zbl 0994.34020

The reversibility is an interesting concept useful in the qualitative theory of differential equations that implies certain geometric properties. For instance, if an orbit of a reversible vector field meets the fixed-point set at two distinct points then it is necessarily a symmetric periodic orbit.

The paper is concerned with the following problem for a class of two-dimensional systems having a center at the singular point: if a system possesses certain symmetries, what can be said about its reversibility? Sufficient and necessary conditions for the analytic planar system to have a center are given and the reversibility of certain classes of polynomial vector fields is discussed.

The paper is concerned with the following problem for a class of two-dimensional systems having a center at the singular point: if a system possesses certain symmetries, what can be said about its reversibility? Sufficient and necessary conditions for the analytic planar system to have a center are given and the reversibility of certain classes of polynomial vector fields is discussed.

Reviewer: Yuri V.Rogovchenko (Famagusta)

### MSC:

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

### Keywords:

center; focus; involution; reversibility; analytic vector field; monodromy conditions; normal forms
PDF
BibTeX
XML
Cite

\textit{M. A. Teixeira} and \textit{J. Yang}, J. Differ. Equations 174, No. 1, 237--251 (2001; Zbl 0994.34020)

Full Text:
DOI

### References:

[1] | Arnold, V.; Il’yashenko, Yu.S., Ordinary differential equations, dynamical systems I, (1988), Springer-Verlag Berlin/New York |

[2] | Belitskii, G., Smooth equivalence of germs of C∞ of vector fields with one zero or a pair of pure imaginary eigenvalues, Funct. anal. appl., 20, 253-259, (1986) · Zbl 0657.58027 |

[3] | Bruno, A.D., Local methods in nonlinear differential equations, (1989), Springer-Verlag Berlin |

[4] | Christorpher, C.J.; Lloyd, N.G., On the paper of jin and Wang concerning the conditions for a center in certain cubic systems, Bull. London math. soc., 5-12, (1990) · Zbl 0696.34026 |

[5] | Christorpher, C.J.; Lloyd, N.G.; Pearson, J.M., On Cherkas’s method for center conditions, Nonlinear world, 2, 459-469, (1995) · Zbl 0833.34023 |

[6] | Lloyd, N.G.; Pearson, J.M., Five limit cycles for a simple cubic system, Publ. mat., 41, 199-208, (1997) · Zbl 0885.34029 |

[7] | Moussu, R., Symmetry and normal form of degenerate centers and foci, Ergodic theory dynam. systems, 2, 241-251, (1982) |

[8] | Montgomery, D.; Zippin, L., Topological transformations groups, (1995), Interscience New York · JFM 66.0959.03 |

[9] | E. Strozyna, and, H. Zoladek, The analytic and formal normal form for the nilpotent singularity, preprint. · Zbl 1005.34034 |

[10] | Takens, F., Singularities of vector fields, Publ. math., inst. hautes etudes sci., 43, 47-100, (1974) · Zbl 0279.58009 |

[11] | J. Yang, Polynomial normal forms of vector fields on R3, Duke Math. J, in press. |

[12] | Zoladek, H., The classification of reversible cubic system with center, Topol. methods nonlinear anal., 4, 79-136, (1994) · Zbl 0820.34016 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.