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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the differentiability of first integrals of two dimensional flows


Authors: Weigu Li, Jaume Llibre, Marcel Nicolau and Xiang Zhang
Journal: Proc. Amer. Math. Soc. 130 (2002), 2079-2088
MSC (2000): Primary 34C05, 34C40, 37C10
Published electronically: January 17, 2002
MathSciNet review: 1896044
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Abstract: By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Giné, and Llibre in 1999. For a given two dimensional flow, what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points $\gamma_i,\,i=1,2,\cdots,l$, such that $\mathbb R^2\backslash\left(\bigcup^{l}_{i=1}\gamma_i\right)$ has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in $\mathbb R^3$, there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in $\mathbb R^3$ is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.


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Additional Information

Weigu Li
Affiliation: Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China
Email: weigu@sxx0.math.pku.edu.cn

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 – Bellaterra, Barcelona, Spain
Email: jllibre@mat.uab.es

Marcel Nicolau
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 – Bellaterra, Barcelona, Spain
Email: nicolau@mat.uab.es

Xiang Zhang
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
Email: xzhang@pine.njnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06310-4
PII: S 0002-9939(02)06310-4
Keywords: Two dimensional flows, polynomial systems, first integrals, differentiability
Received by editor(s): August 4, 2000
Received by editor(s) in revised form: February 16, 2001
Published electronically: January 17, 2002
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society