Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T09:35:14.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal forms for almost periodic differential systems

Published online by Cambridge University Press:  01 April 2009

WEIGU LI ,
JAUME LLIBRE  and
HAO WU
WEIGU LI
Affiliation:
School of Mathematical Sciences, Peking University, 100871 Beijing, China (email: [email protected])
JAUME LLIBRE
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (email: [email protected])
HAO WU
Affiliation:
School of Mathematical Sciences, Peking University, 100871 Beijing, China (email: [email protected]) Department of Mathematics, Southeast University, 210096 Nanjing, Jiangsu, China (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we prove smooth conjugate theorems of Sternberg type for almost periodic differential systems, based on the Lyapunov exponents of the corresponding reduced systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 637 - 656
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, L.. Random Dynamical Systems (Springer Monographs in Mathematics). Springer, Berlin, 1998.Google Scholar
[2]Bainov, D. and Simenov, P.. Integral Inequalities and Applications. Kluwer, Dordrecht, 1992.CrossRefGoogle Scholar
[3]Bibikov, Y. N.. Local Theory of Nonlinear Analytic Ordinary Differential Equations (Lecture Notes in Mathematics, 702). Springer, Berlin, 1979.CrossRefGoogle Scholar
[4]Broer, H. W., Ciocci, M. C. and Hanßmann, H.. The quasi-periodic reversible Hopf bifurcation. Internat. J. Bifur. Chaos 17 (2007), 2605–2623.CrossRefGoogle Scholar
[5]Broer, H. W., Hanßmann, H. and Hoo, J.. The quasi-periodic Hamiltonian Hopf bifurcation. Nonlinearity 20 (2007), 417–460.CrossRefGoogle Scholar
[6]Broer, H. W., Hoo, J. and Naudot, V.. Normal linear stability of quasi-periodic tori. J. Differential Equations 232 (2007), 355–418.CrossRefGoogle Scholar
[7]Broer, H. W., Huitema, G. B., Takens, F. and Braaksma, B. L. J.. Unfoldings and bifurcations of quasi-periodic tori. Mem. Amer. Math. Soc. 421(83) (1990), 175.Google Scholar
[8]Chen, K. T.. Equivalence and decomposition of vector fields about an elementary critical point. Amer. J. Math. 85 (1963), 693–722.Google Scholar
[9]Chow, S. N. and Lu, K.. Invariant manifolds and foliations for quasiperiodic systems. J. Differential Equations 117 (1995), 1–27.CrossRefGoogle Scholar
[10]Coppel, W. A.. Dichotomies in Stability Theory (Lecture Notes in Mathematics, 629). Springer, Berlin, 1978.CrossRefGoogle Scholar
[11]Fink, A. M.. Almost Periodic Differential Equations (Lecture Notes in Mathematics, 377). Springer, Berlin, 1974.CrossRefGoogle Scholar
[12]Hale, J. K.. Ordinary Differential Equations, 2nd edn. Robert E. Krieger Publishing Co., Huntington NY, 1980.Google Scholar
[13]Il’yashenko, Yu. S. and Yakovenko, S. Yu.. Finitely-smooth normal forms of local families of diffeomorphisms and vector fields. Russian Math. Surveys 46 (1991), 1–43.CrossRefGoogle Scholar
[14]Li, W. and Lu, K.. Poincaré theorems for random dynamical systems. Ergod. Th. & Dynam. Sys. 25 (2005), 1221–1236.CrossRefGoogle Scholar
[15]Li, W. and Lu, K.. Sternberg theorems for random dynamical systems. Comm. Pure Appl. Math. 58 (2005), 941–988.CrossRefGoogle Scholar
[16]Li, W., Llibre, J. and Wu, H.. Polynomial and linearized normal forms for almost periodic difference systems. J. Difference Equ. Appl. to appear.Google Scholar
[17]Mitropolsky, Yu. A.. The averaging method and the problem on separation of variables. Differential Equations, Dynamical Systems, and Control Science (Lecture Notes in Pure and Applied Mathematics, 152). Dekker, New York, 1994, pp. 213–223.Google Scholar
[18]Palmer, K. J.. On the reducibility of almost periodic systems of linear differential equations. J. Differential Equations 36 (1980), 374–390.CrossRefGoogle Scholar
[19]Poincaré, H.. Mémoire sur les courbes définies par une équation différentielle. Thesis, 1879. Also Oeuvres I, 59–129, Gauthier Villars, Paris, 1928.Google Scholar
[20]Sacker, R. J. and Sell, G. R.. A spectral theory for linear differential systems. J. Differential Equations 27 (1978), 320–358.CrossRefGoogle Scholar
[21]Sanders, J. A. and Verhulst, F.. Averaging Methods in Nonlinear Dynamical Systems (Applied Mathematical Sciences, 59). Springer, New York, 1985.CrossRefGoogle Scholar
[22]Sevryuk, M. B.. The comment to problem 1970–1. Arnold’s Problems. Ed. V. I. Arnold. Springer, Berlin, 2004, pp. 226–231.Google Scholar
[23]Siegmund, S.. Normal forms for nonautonomous difference equations. Advances in difference equations, IV. Comput. Math. Appl. 45 (2003), 1059–1073.CrossRefGoogle Scholar
[24]Sternberg, S.. Finite Lie groups and the formal aspects of dynamical systems. J. Math. Mech. 10 (1961), 451–474.Google Scholar
[25]Takens, F.. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier 23 (1973), 163–195.CrossRefGoogle Scholar
[26]Vanderbauwhede, A.. Center manifolds and their basic properties. An introduction. Delft Progr. Rep. 12 (1988), 57–78.Google Scholar
[27]Zhang, W.. Generalized exponential dichotomies and invariant manifolds for differential equations. Adv. Math. (China) 22 (1993), 1–45.Google Scholar