### Abstract

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f0 (x, y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f0 (x, y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f0 (x, y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles. © 2007 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 375-398 |

Journal | Journal of Differential Equations |

Volume | 240 |

DOIs | |

Publication status | Published - 15 Sep 2007 |

### Keywords

- Bifurcation
- Invariant algebraic curve
- Multiple limit cycle
- Planar vector field
- Stability
- m-Solution

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## Cite this

Gasull, A., Giné, J., & Grau, M. (2007). Multiplicity of limit cycles and analytic m-solutions for planar differential systems.

*Journal of Differential Equations*,*240*, 375-398. https://doi.org/10.1016/j.jde.2007.05.037