On the differentiability of first integrals of two dimensional flows |
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Authors: | Weigu Li Jaume Llibre Marcel Nicolau Xiang Zhang |
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Affiliation: | Department of Mathematics, Peking University, Beijing 100871, People's Republic of China ; Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 -- Bellaterra, Barcelona, Spain ; Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 -- Bellaterra, Barcelona, Spain ; Department of Mathematics, Nanjing Normal University, Nanjing 210097, People's Republic of China |
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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 , such that 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 , there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in 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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Keywords: | Two dimensional flows polynomial systems first integrals differentiability |
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