Darboux theory of integrability in \(\mathbb C^n\) taking into account the multiplicity

DOI10.1016/j.jde.2008.07.020zbMath1163.34003OpenAlexW2054702357MaRDI QIDQ2378206

Jaume Llibre, Xiang Zhang

Publication date: 7 January 2009

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jde.2008.07.020

zbMATH Keywords

multiplicitypolynomial vector fieldsDarboux first integralsrational first integralsinvariant algebraic hypersurfaces

Mathematics Subject Classification ID

Symmetries, invariants of ordinary differential equations (34C14) Explicit solutions, first integrals of ordinary differential equations (34A05) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Invariant manifolds for ordinary differential equations (34C45)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• A family of isochronous foci with Darboux first integral
• Multiplicity of invariant algebraic curves in polynomial vector fields
• Equations de Pfaff algébriques
• M. N. Lagutinskij (1871-1915): A misunderstood mathematician
• Configurations of limit cycles and planar polynomial vector fields.
• Rational first integrals in the Darboux theory of integrability in \(\mathbb C^n\)
• Planar quadratic vector fields with invariant lines of total multiplicity at least five
• On the integrability of the Einstein-Yang-Mills equations
• Algebraic Particular Integrals, Integrability and the Problem of the Center
• Rikitake system: analytic and Darbouxian integrals
• Integrability of the Bianchi IX system
• Exponential factors and Darbouxian first integrals of the Lorenz system
• Vector fields, invariant varieties and linear systems.

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