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Survey on Integrability and Center Problem of Nilpotent Planar Differential Systems

    1. [1] Universidad de Huelva

      Universidad de Huelva

      Huelva, España

    2. [2] Universitat de Lleida

      Universitat de Lleida

      Lérida, España

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
  • Idioma: inglés
  • DOI: 10.1007/s12346-026-01552-7
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  • Resumen

    • We overview the studies on integrability, the center problem and the existence of inverse integrating factor for a planar differential system with a nilpotent singularity.

      We analyze the connections among these three concepts and provide several examples to enhance understanding of the topics discussed.

      We consider the planar differential systems ˙x = F(x) where F = (P, Q)T is a vector field defined in a domain U ⊂ C2 whose components P and Q are analytic in a neighborhood of the origin and assume that the origin is an isolated singular point. We recall a classification of the singular points of a vector field according to the expression of DF(0), the linear part of a vector field valued at the origin. We say that the origin is an elementary singular point if at least one eigenvalue of DF(0) is non-zero. Otherwise, both eigenvalues are zero, the origin is a degenerate singular point.

      In this paper, we deal with differential systems whose origin is degenerate but DF(0) is non-zero, that is, DF(0) is a nilpotent matrix. For that reason, these systems are called nilpotent differential systems. Any nilpotent differential system, by means of a linear change of variables, is transformed into a differential system ˙ x = y + ¯P (x, y), ˙y = ¯Q (x, y), (1) where ¯P , ¯Q are analytic functions without constant and linear terms. In this case, we say that the origin is a nilpotent singularity, also-called a Takens-Bogdanov-Arnold singularity due to they were the first authors that studied it, see [37, 39, 40, 133, 134].

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