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Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

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Abstract

A quadratic polynomial differential systemcan be identified with a single point of ℝ12 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 ℝ12 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.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Joan C. Artés & Jaume Llibre

  2. Institute of Mathematics and Computer Science, Academy of Science of Moldova, 5 Academiei str, Chişinău, MD-2028, Moldova

    Nicolae Vulpe

Authors
  1. Joan C. Artés
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  2. Jaume Llibre
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  3. Nicolae Vulpe
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Corresponding author

Correspondence to Jaume Llibre.

Additional information

The first two authors have been supported by the grants MCD/FEDER number MTM2008-03437 and CIRIT 2009SGR410. The third author has been partially supported by CRDFMRDA CERIM-1006-06.

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Artés, J.C., Llibre, J. & Vulpe, N. Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12 . Rend. Circ. Mat. Palermo 59, 419–449 (2010). https://doi.org/10.1007/s12215-010-0032-0

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