Fixed point indices and invariant periodic sets of holomorphic systems

Zhang, Guang

doi:10.1090/S0002-9939-06-08821-6

Fixed point indices and invariant periodic sets of holomorphic systems
HTML articles powered by AMS MathViewer

by Guang Yuan Zhang PDF

Proc. Amer. Math. Soc. 135 (2007), 767-776 Request permission

Abstract:

This note presents an intuitive method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham’s theorem in a more general version.

References

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
E. M. Chirka, Complex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by R. A. M. Hoksbergen. MR 1111477, DOI 10.1007/978-94-009-2366-9
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
Gregor, J., Dynamical systems with regular hand-side, Pokroky Mat. Fys. Astronom. 3 (1958), 153–160 (Zbl 081.30802).
Otomar Hájek, Notes on meromorphic dynamical systems. I, Czechoslovak Math. J. 16(91) (1966), 14–27 (English, with Russian summary). MR 194661
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, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
D. J. Needham, A centre theorem for two-dimensional complex holomorphic systems and its generalization, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1939, 225–232. MR 1349506, DOI 10.1098/rspa.1995.0082
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
M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189–191. MR 350782, DOI 10.1016/0040-9383(74)90009-3
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
Guangyuan Zhang, Multiplicities of fixed points of holomorphic maps in several complex variables, Sci. China Ser. A 44 (2001), no. 3, 330–337. MR 1828756, DOI 10.1007/BF02878713
Guang Yuan Zhang, Bifurcations of periodic points of holomorphic maps from $\mathbf C^2$ into $\mathbf C^2$, Proc. London Math. Soc. (3) 79 (1999), no. 2, 353–380. MR 1702246, DOI 10.1112/S0024611599011910