From an iteration formula to Poincaré’s Isochronous Center Theorem for holomorphic vector fields
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Abstract:
We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincaré, for the condition under which a singularity of a holomorphic vector field is an isochronous center.References
- V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York, 1988. Translated from the Russian by Joseph Szücs [József M. Szűcs]. MR 947141, DOI 10.1007/978-1-4612-1037-5
- D. K. Arrowsmith and C. M. Place, An introduction to dynamical systems, Cambridge University Press, Cambridge, 1990. MR 1069752
- Louis Brickman and E. S. Thomas, Conformal equivalence of analytic flows, J. Differential Equations 25 (1977), no. 3, 310–324. MR 447674, DOI 10.1016/0022-0396(77)90047-X
- Javier Chavarriga and Marco Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999), no. 1, 1–70. MR 1747197, DOI 10.1007/BF02969404
- L. A. Cherkas, V. G. Romanovskii, and H. Żoła̧dek, The centre conditions for a certain cubic system, Differential Equations Dynam. Systems 5 (1997), no. 3-4, 299–302. Planar nonlinear dynamical systems (Delft, 1995). MR 1660202
- C. J. Christopher and J. Devlin, Isochronous centers in planar polynomial systems, SIAM J. Math. Anal. 28 (1997), no. 1, 162–177. MR 1427732, DOI 10.1137/S0036141093259245
- Lenore Feigenbaum, The center of oscillation versus the textbook writers of the early 18th century, From ancient omens to statistical mechanics, Acta Hist. Sci. Nat. Med. Edidit Bibl. Univ. Haun., vol. 39, Univ. Lib. Copenhagen, Copenhagen, 1987, pp. 193–202. MR 961883
- J.-P. Francoise, Isochronous systems and perturbation theory, J. Nonlinear Math. Phys. 12 (2005), no. suppl. 1, 315–326. MR 2117189, DOI 10.2991/jnmp.2005.12.s1.25
- Gregor, J., Dynamical systems with regular hand-side, Pokroky Mat. Fys. Astronom. 3 (1958), 153–160.
- Otomar Hájek, Notes on meromorphic dynamical systems. I, Czechoslovak Math. J. 16(91) (1966), 14–27 (English, with Russian summary). MR 194661, DOI 10.21136/CMJ.1966.100705
- N. A. Lukaševič, The isochronism of a center of certain systems of differential equations, Differencial′nye Uravnenija 1 (1965), 295–302 (Russian). MR 0197863
- John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
- D. J. Needham and S. McAllister, Centre families in two-dimensional complex holomorphic dynamical systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1976, 2267–2278. MR 1639872, DOI 10.1098/rspa.1998.0258
- Marco Paluszny, On periodic solutions of polynomial ODEs in the plane, J. Differential Equations 53 (1984), no. 1, 24–29. MR 747404, DOI 10.1016/0022-0396(84)90023-8
- M. Sabatini, Dynamics of commuting systems on two-dimensional manifolds, Ann. Mat. Pura Appl. (4) 173 (1997), 213–232. MR 1625543, DOI 10.1007/BF01783469
- Massimo Villarini, Regularity properties of the period function near a center of a planar vector field, Nonlinear Anal. 19 (1992), no. 8, 787–803. MR 1186791, DOI 10.1016/0362-546X(92)90222-Z
- Zhang, G. Y., Fixed Point Indices and Invariant Periodic Sets of Holomorphic Systems, to appear in Proc. Amer. Math. Soc.
Additional Information
- Guang Yuan Zhang
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
- Email: gyzhang@math.tsinghua.edu.cn
- Received by editor(s): November 23, 2005
- Received by editor(s) in revised form: May 30, 2006
- Published electronically: May 8, 2007
- Additional Notes: The author is supported by Chinese NSFC 10271063 and 10571009
- Communicated by: Carmen C. Chicone
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2887-2891
- MSC (2000): Primary 32H50, 32M25, 37C27
- DOI: https://doi.org/10.1090/S0002-9939-07-08802-8
- MathSciNet review: 2317965