Normal form and linearization for quasiperiodic systems
HTML articles powered by AMS MathViewer
- by Shui-Nee Chow, Kening Lu and Yun Qiu Shen
- Trans. Amer. Math. Soc. 331 (1992), 361-376
- DOI: https://doi.org/10.1090/S0002-9947-1992-1076612-1
- PDF | Request permission
Abstract:
In this paper, we consider the following system of differential equations: \[ \dot \theta = \omega + \Theta (\theta ,z), \quad \dot z = Az + f(\theta ,z),\] where $\theta \in {C^m}$, $\omega = ({\omega _1}, \ldots ,{\omega _m}) \in {R^m}$, $z \in {C^n}$, $A$ is a diagonalizable matrix, $f$ and $\Theta$ are analytic functions in both variables and $2\pi$-periodic in each component of the vector $\theta ,\Theta = O(|z|)$ and $f = O(|z{|^2})$ as $z \to 0$. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations \[ \dot \theta = \omega , \quad \dot z = Az\] by an analytic transformation provided that the eigenvalues of $A$ and the frequency $\omega$ satisfy certain small-divisor conditions.References
- V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786
- È. G. Belaga, The reducibility of a system of ordinary differential equations in the neighborhood of a conditionally periodic motion, Dokl. Akad. Nauk SSSR 143 (1962), 255–258 (Russian). MR 0138845 B. L. J. Braaksma and H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. Henri Poincaré4 (1987), 115-168.
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
- K. R. Meyer, The implicit function theorem and analytic differential equations, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 191–208. MR 0650636
- Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 208078, DOI 10.1007/BF01399536
- Helmut Rüssmann, Kleine Nenner. II. Bemerkungen zur Newtonschen Methode, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1972), 1–10 (German). MR 309297
- Eduard Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 855–866. MR 0461575
- H. W. Broer, G. B. Huitema, F. Takens, and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (1990), no. 421, viii+175. MR 1041003, DOI 10.1090/memo/0421
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 331 (1992), 361-376
- MSC: Primary 34C20; Secondary 58F36, 70H05
- DOI: https://doi.org/10.1090/S0002-9947-1992-1076612-1
- MathSciNet review: 1076612