The continuity of reduction of hypersurfaces in \(\mathbb{C}^ 2\) to a normal form. (English. Russian original) Zbl 0813.32016
Funct. Anal. Appl. 27, No. 4, 288-290 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 81-84 (1993).
Let \(M\) be a real analytic hypersurface in \(\mathbb{C}^ 2\) given by an equation \(v=F(z,u,y)\) with complex coordinates \(z=x+iy\), \(w=u+iv\). Suppose that the Levi form of \(M\) is nondegenerate at some point. After a plane normalization given in the author’s previous paper [Math. Notes 52, No. 1, 687-694 (1992); translation from Mat. Zametki 52, No. 1, 76-86 (1992; Zbl 0769.32007)], we obtain a new equation for \(M:F(x,u,y) = 2y + \sum_{k \geq 4} F_ k (x,u)y^ k\), where coefficients \(F_ 4(x,u)\) and \(F_ 5(x,u)\) satisfy some additional conditions. Let \(\{M_ \alpha\}\) be a family of hypersurfaces \(M_ \alpha\) depending continuously on a real parameter \(\alpha\). The author considers the reduction of \(M_ \alpha\) to a plane normal form and obtains the conditions under which the normal equations and normalizing mappings depend on \(\alpha\) continuously.
Reviewer: A.Morimoto (Nagoya)
MSC:
| 32V40 | Real submanifolds in complex manifolds |
| 32C05 | Real-analytic manifolds, real-analytic spaces |
References:
| [1] | V. I. Arnold, ”Remarks on perturbation theory for problems of Mathieu type,” Usp. Mat. Nauk,38, No. 4, 189–203 (1983). |
| [2] | V. I. Arnold, ”Small denominators I. Mappings of the circumference onto itself,” Trans. Amer. Math. Soc.,46, 213–284 (1965). · Zbl 0152.41905 |
| [3] | O. G. Galkin, ”Phase-locking for Mathieu-type vector fields on the torus,” Funkts. Anal. Prilozhen.,26, No. 1, 1–8 (1992). · Zbl 0828.20025 · doi:10.1007/BF01077066 |
| [4] | A. Khinchin, Continued fractions, Groningen, Nordhorff (1963). · Zbl 0117.28601 |
| [5] | C. Baesens, J. Guckenheimer, S. Kim, and R. S. MacKay, ”Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,” Phys. D,49, 387–475 (1991). · Zbl 0734.58036 · doi:10.1016/0167-2789(91)90155-3 |
| [6] | R. E. Ecke, J. D. Farmer, and D. K. Umberger, ”Scaling of the Arnold tongues,” Nonlinearity,2, 175–196 (1989). · Zbl 0689.58017 · doi:10.1088/0951-7715/2/2/001 |
| [7] | J. Franks and M. Misiurewicz, ”Rotation sets of toral flows,” Proc. Amer. Math. Soc.,109, 243–249 (1990). · Zbl 0701.57016 · doi:10.1090/S0002-9939-1990-1021217-5 |
| [8] | O. G. Galkin, ”Resonance regions for Mathieu type dynamical systems on a torus,” Phys. D,39, 287–298 (1989). · Zbl 0695.58025 · doi:10.1016/0167-2789(89)90011-0 |
| [9] | C. Grebogi, E. Ott, and J. A. Yorke, ”Attractors on ann-torus: quasiperiodicity versus chaos,” Phys. D,15, 354–373 (1985). · Zbl 0577.58023 · doi:10.1016/S0167-2789(85)80004-X |
| [10] | G. R. Hall, ”Resonance zones in two-parameter families of circle homeomorphisms,” SIAM J. Math. Anal.,15, 1075–1081 (1984). · Zbl 0554.58040 · doi:10.1137/0515083 |
| [11] | S. Kim, R. S. MacKay, and J. Guckenheimer, ”Resonance regions for families of torus maps,” Nonlinearity,2, 391–404 (1989). · Zbl 0678.58034 · doi:10.1088/0951-7715/2/3/001 |
| [12] | M. Misiurewicz and K. Ziemian, ”Rotation sets for maps of tori,” J. London Math. Soc.,40, 490–506 (1989). · Zbl 0663.58022 · doi:10.1112/jlms/s2-40.3.490 |
| [13] | S. Newhouse, J. Palis, and F. Takens, ”Bifurcations and stability of families of diffeomorphisms,” Publ. Math. IHES,57, 5–72 (1983). · Zbl 0518.58031 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
