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Normal Forms for Polynomial Differential Systems in ℝ³ Having an Invariant Quadric and a Darboux Invariant

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, UNESP – Univ Estadual Paulista, 19060-900 Presidente Prudente, São Paulo, Brazil

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Alisson de Carvalho Reinol

Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, UNESP – Univ Estadual Paulista, 19060-900 Presidente Prudente, São Paulo, Brazil

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https://doi.org/10.1142/S0218127415500157Cited by:8 (Source: Crossref)

Abstract

We give the normal forms of all polynomial differential systems in ℝ³ which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example, we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric.

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