Quadratic systems with a polynomial first integral: A complete classification in the coefficient space R12

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Abstract

In this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non-degenerate quadratic system in order to determine if it has or not a polynomial first integral without using any normal form. We obtain that the existence of polynomial first integral is directly related with the fact that all the roots of a convenient cubic polynomial are rational and negative. The coefficients of this cubic polynomial are invariants related with some geometric properties of the system. © 2008 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)3535-3558
JournalJournal of Differential Equations
Volume246
DOIs
Publication statusPublished - 1 May 2009

Keywords

  • Affine invariant polynomial
  • Integrability
  • Polynomial first integral
  • Quadratic vector fields

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