Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T10:48:01.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-periodic motions in strongly dissipative forced systems

Published online by Cambridge University Press:  03 August 2009

GUIDO GENTILE
GUIDO GENTILE*
Affiliation:
Dipartimento di Matematica, Università di Roma Tre, Roma, I-00146, Italy (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1457 - 1469
Copyright
Copyright © Cambridge University Press 2009

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]Bartuccelli, M. V., Deane, J. H. B. and Gentile, G.. Globally and locally attractive solutions for quasi-periodically forced systems. J. Math. Anal. Appl. 328(1) (2007), 699–714.CrossRefGoogle Scholar
[2]Bartuccelli, M. V., Deane, J. H. B., Gentile, G. and Marsh, L.. Invariant sets for the varactor equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462(2066) (2006), 439–457.Google Scholar
[3]Bollobás, B.. Graph Theory. An Introductory Course (Graduate Texts in Mathematics, 63). Springer, New York, 1979.Google Scholar
[4]Broer, H. W., Huitema, G. B. and Sevryuk, M. B.. Quasi-Periodic Motions in Families of Dynamical Systems. Order Amidst Chaos (Lecture Notes in Mathematics, 1645). Springer, Berlin, 1996.Google Scholar
[5]Bryuno, A. D.. Analytic form of differential equations. I. Trans. Moscow Math. Soc. 25 (1971), 131–288; II. Trans. Moscow Math. Soc. 26 (1972), 199–239.Google Scholar
[6]Corsi, L. and Gentile, G.. Melnikov theory to all orders and Puiseux series for subharmonic solutions. J. Math. Phys. 49(11) (2008), 112701-1–29.CrossRefGoogle Scholar
[7]Costin, O., Gallavotti, G., Gentile, G. and Giuliani, A.. Borel summability and Lindstedt series. Comm. Math. Phys. 269(1) (2007), 175–193.CrossRefGoogle Scholar
[8]Gallavotti, G., Bonetto, F. and Gentile, G.. Aspects of Ergodic, Qualitative and Statistical Theory of Motion (Texts and Monographs in Physics). Springer, Berlin, 2004.CrossRefGoogle Scholar
[9]Gentile, G., Bartuccelli, M. V. and Deane, J. H. B.. Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms. J. Math. Phys. 46(6) (2005), 062704-1–21.CrossRefGoogle Scholar
[10]Gentile, G., Bartuccelli, M. V. and Deane, J. H. B.. Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J. Math. Phys. 47(7) (2006), 072702-1–10.CrossRefGoogle Scholar
[11]Gentile, G.. Quasi-periodic solutions for two-level systems. Comm. Math. Phys. 242(1–2) (2003), 221–250.CrossRefGoogle Scholar
[12]Gentile, G.. Resummation of perturbation series and reducibility for Bryuno skew-product flows. J. Stat. Phys. 125(2) (2006), 321–361.CrossRefGoogle Scholar
[13]Gentile, G.. Degenerate lower-dimensional tori under the Bryuno condition. Ergod. Th. & Dynam. Sys. 27(2) (2007), 427–457.CrossRefGoogle Scholar
[14]Harary, F.. Graph Theory. Addison-Wesley, Reading, Mass, 1969.CrossRefGoogle Scholar
[15]Harary, F. and Palmer, E. M.. Graphical Enumeration. Academic Press, New York, 1973.Google Scholar
[16]Lopes Dias, J.. A normal form theorem for Brjuno skew systems through renormalization. J. Differential Equations 230(1) (2006), 1–23.CrossRefGoogle Scholar
[17]Marmi, S., Moussa, P. and Yoccoz, J.-Ch.. Some properties of real and complex Brjuno functions. Frontiers in Number Theory, Physics, and Geometry. I. Eds. Cartier, P., Julia, B., Moussa, P. and Vanhove, P.. Springer, Berlin, 2006, pp. 601–623.Google Scholar
[18]Sokal, A. D.. An improvement of Watson’s theorem on Borel summability. J. Math. Phys. 21(2) (1980), 261–263.CrossRefGoogle Scholar
[19]Ponce, M.. On the persistence of invariant curves for fibered holomorphic transformations. Comm. Math. Phys. 289(1) (2009), 1–44.CrossRefGoogle Scholar
[20]Stoker, J. J.. Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers, New York, NY, 1950.Google Scholar