### Abstract

We give an upper bound of the index of an isolated equilibrium point of a C 1 vector field in the plane. The vector field is decomposed in gradient and Hamiltonian components. This decomposition is related with the Loewner vector field. Associated to this decomposition we consider the set Π where the gradient and Hamiltonian components are linearly dependent. The number of branches of Π starting at the equilibrium point determines the upper bound of the index. © 2012 Elsevier Inc.

Original language | English |
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Pages (from-to) | 2460-2473 |

Journal | Journal of Differential Equations |

Volume | 253 |

Issue number | 8 |

DOIs | |

Publication status | Published - 15 Oct 2012 |

### Keywords

- Gradient systems
- Hamiltonian systems
- Index
- Planar differential systems

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## Cite this

Llibre, J., & Martínez-Alfaro, J. (2012). An upper bound of the index of an equilibrium point in the plane.

*Journal of Differential Equations*,*253*(8), 2460-2473. https://doi.org/10.1016/j.jde.2012.07.001