Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Normal form and linearization for quasiperiodic systems


Authors: Shui-Nee Chow, Kening Lu and Yun Qiu Shen
Journal: Trans. Amer. Math. Soc. 331 (1992), 361-376
MSC: Primary 34C20; Secondary 58F36, 70H05
MathSciNet review: 1076612
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the following system of differential equations:

$\displaystyle \dot \theta = \omega + \Theta (\theta,z), \quad \dot z = Az + f(\theta,z),$

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(\vert z\vert)$ and $ f = O(\vert z{\vert^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

$\displaystyle \dot \theta = \omega, \quad \dot z = Az$

by an analytic transformation provided that the eigenvalues of $ A$ and the frequency $ \omega $ satisfy certain small-divisor conditions.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786 (84d:58023)
  • [2] 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 (25 #2286)
  • [3] B. L. J. Braaksma and H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. Henri Poincaré4 (1987), 115-168.
  • [4] Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 251, Springer-Verlag, New York, 1982. MR 660633 (84e:58019)
  • [5] 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 (85f:58002)
  • [6] 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), Springer, Berlin, 1975, pp. 191–208. Lecture Notes in Math., Vol. 468. MR 0650636 (58 #31247)
  • [7] Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 0208078 (34 #7888)
  • [8] Helmut Rüssmann, Kleine Nenner. II. Bemerkungen zur Newtonschen Methode, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1972), 1–10 (German). MR 0309297 (46 #8407)
  • [9] 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), Springer, Berlin, 1977, pp. 855–866. Lecture Notes in Math., Vol. 597. MR 0461575 (57 #1560)
  • [10] 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 (91e:58156)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34C20, 58F36, 70H05

Retrieve articles in all journals with MSC: 34C20, 58F36, 70H05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1076612-1
PII: S 0002-9947(1992)1076612-1
Article copyright: © Copyright 1992 American Mathematical Society