On the differentiability of first integrals of two dimensional flows

Li, Weigu; Llibre, Jaume; Nicolau, Marcel; Zhang, Xiang

doi:10.1090/S0002-9939-02-06310-4

On the differentiability of first integrals of two dimensional flows
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by Weigu Li, Jaume Llibre, Marcel Nicolau and Xiang Zhang PDF

Proc. Amer. Math. Soc. 130 (2002), 2079-2088 Request permission

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.

References

D. V. Anosov and V. I. Arnol′d (eds.), Dynamical systems. I, Encyclopaedia of Mathematical Sciences, vol. 1, Springer-Verlag, Berlin, 1988. Ordinary differential equations and smooth dynamical systems; Translated from the Russian [ MR0823488 (86i:58037)]. MR 970793, DOI 10.1007/978-3-642-61551-1
M. Izabel T. Camacho, Geometric properties of homogeneous vector fields of degree two in $\textbf {R}^{3}$, Trans. Amer. Math. Soc. 268 (1981), no. 1, 79–101. MR 628447, DOI 10.1090/S0002-9947-1981-0628447-1
Javier Chavarriga, Hector Giacomini, Jaume Giné, and Jaume Llibre, On the integrability of two-dimensional flows, J. Differential Equations 157 (1999), no. 1, 163–182. MR 1710019, DOI 10.1006/jdeq.1998.3621
Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
Jean Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1992 (French, with French summary). MR 1399559
Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0486784
William Huebsch and Marston Morse, Diffeomorphisms of manifolds, Rend. Circ. Mat. Palermo (2) 11 (1962), 291–318. MR 172297, DOI 10.1007/BF02843877
Yu. S. Il′yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol. 94, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by H. H. McFaden. MR 1133882, DOI 10.1090/mmono/094
Solomon Lefschetz, Differential equations: geometric theory, 2nd ed., Dover Publications, Inc., New York, 1977. MR 0435481
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Dean A. Neumann, Classification of continuous flows on $2$-manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. MR 356138, DOI 10.1090/S0002-9939-1975-0356138-6
Robert Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014, DOI 10.1007/978-3-0348-8798-4
H. L. Royden, The analytic approximation of differentiable mappings, Math. Ann. 139 (1960), 171–179 (1960). MR 113229, DOI 10.1007/BF01352908
Stephen Schecter and Michael F. Singer, Planar polynomial foliations, Proc. Amer. Math. Soc. 79 (1980), no. 4, 649–656. MR 572321, DOI 10.1090/S0002-9939-1980-0572321-0
Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0295244
H. Whitney, Differentiable manifolds, Annals of Math. 37 (1936), 645–680.
Ye Yanqian, Qualitative Theory of polynomial Differential Systems (in Chinese), Shanghai Scientific $\&$ Technical Publishers, Shanghai, 1995.