Quadratic systems with a rational first integral of degree three: A complete classification in the coefficient space ℝ 12

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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 languageEnglish
Pages (from-to)419-449
JournalRendiconti del Circolo Matematico di Palermo
Volume59
Issue number3
DOIs
Publication statusPublished - 1 Dec 2010

Keywords

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

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