Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in ℂ2

Abstract

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane [Formula: see text]. The criterion uses solutions of the form [Formula: see text] of the differential system in the plane and their associated cofactors, where [Formula: see text] is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in [Formula: see text] would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability.

On the Dynamics of Higgins–Selkov, Selkov and Brusellator Oscillators

“…Currently, the study of new integrability theories has been the objective of several recently works; see [1][2][3][4][5]. The classical Darboux theory of integrability plays a central role in the integrability of the polynomial differential systems.…”

“…The alternative method to control the existence of Darboux polynomials is based on computing the solutions of the associated differential equation in terms of Puiseux series and study their existence and number; see [1,[7][8][9][10] and references therein. The case of formal series solutions was studied in [3,4], called the Weierstrass integrability. The Weierstrass and Puiseux integrability methods have been applied to Liénard systems [11] and in the context of the Jacobian conjecture [12].…”

Non-algebraic limit cycles in Holling type III zooplankton-phytoplankton models

“…For related works which also apply formal and Puiseux series to planar polynomial systems see [4,7,8].…”