Darboux theory of integrability in Cn taking into account the multiplicity

By Jaume Llibre, Xiang Zhang. 2009

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Abstract Darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces, called now the Darboux theory of integrability. In 1979 Jouanolou successfully improved the Darboux theory of integrability characterizing the existence of rational first integrals, for this he used sophisticated tools of algebraic geometry. The aim of this paper is to improve the classical result of Darboux and the new one of Jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces. Additionally our proof of the improved result of Jouanolou is much simpler and elementary than the original one. Some examples show that the improved Darboux theory of integrability with multiplicity is much useful than the classical one.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/801c17a5e3a1add8813f6406fb8bbced72aa52fc

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Citations in the corpus

Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems (2011, methodology): “Many different methods have been used for studying the existence of first integrals of non-linear differential systems based on Noether symmetries [6], the Darboux theory of integrability [8, 17 ], the Lie symmetries [1,23], the Painlev´e analysis [3], the use of Lax pairs [12], the direct method [9”
Global Structure of Quaternion Polynomial Differential Equations (2014, background): “For more information on the Darboux theory of integrability, see for instance [24, 28].”
On the Darboux Integrability of Polynomial Differential Systems (2012, background): “This characterization due to Llibre and Zhang [28] extends the algebraic multiplicity introduced by Christopher, Llibre and Pereira in [11] for invariant algebraic curves of C2 to invariant algebraic hypersurfaces of Cn.”
A Survey on the Inverse Integrating Factor (2009, methodology): “ltiplicity of the invariant algebraic hypersurfaces of a polynomial vector ﬁeld in Cn, the results of Darboux integrability theory of Theorem 3 have been generalized to systems in Cn, where n≥ 2, see [82] and the references therein. An improvement of the previous Darboux theorem is presented in [18]”