### Abstract

We consider the one-parameter family of planar quintic systems, x ̇=y3-x3, y ̇=-x+my5, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0.36, 0.6). In this paper, using the Bendixson-Dulac theorem, we give a new unified proof of all the previous results. We shrink this interval to (0.547, 0.6) and we prove the hyperbolicity of the limit cycle. Furthermore, we consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally, we answer an open question about the change of stability of the origin for an extension of the above systems. © 2013 Elsevier Inc.

Original language | English |
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Pages (from-to) | 321-342 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 413 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 May 2014 |

### Keywords

- Basin of attraction
- Bifurcation
- Dulac function
- Nilpotent point
- Phase portrait on the Poincaré disc
- Planar polynomial system
- Polycycle
- Stability
- Uniqueness and hyperbolicity of the limit cycle

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## Cite this

García-Saldaña, J. D., Gasull, A., & Giacomini, H. (2014). Bifurcation diagram and stability for a one-parameter family of planar vector fields.

*Journal of Mathematical Analysis and Applications*,*413*(1), 321-342. https://doi.org/10.1016/j.jmaa.2013.11.047