### Abstract

A quadratic polynomial differential systemcan be identified with a single point of ℝ 12 through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ℝ 12 having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral. © 2010 Springer-Verlag Italia.

Original language | English |
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Pages (from-to) | 419-449 |

Journal | Rendiconti del Circolo Matematico di Palermo |

Volume | 59 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Dec 2010 |

### Keywords

- Integrability
- Phase portraits
- Quadratic vector fields
- Rational first integral

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## Cite this

Artés, J. C., Llibre, J., & Vulpe, N. (2010). Quadratic systems with a rational first integral of degree three: A complete classification in the coefficient space ℝ

^{12}.*Rendiconti del Circolo Matematico di Palermo*,*59*(3), 419-449. https://doi.org/10.1007/s12215-010-0032-0