Quasiperiodic Solutions of Differential-Difference Equations on a Torus

Abstract

This paper is dedicated to Professor V. A. Pliss on his 70th birthday. In connection with problems considered here, one should note the famous result of V. A. Pliss concerning the structural stability of differential equations on a torus. The first part of the present paper is devoted to the extension to differential-difference equations on an m-dimensional torus of the V. I. Arnold's result [1] about the reducibility of analytic systems on an m-dimensional torus to pure rotation. This result of Arnold was first extended to smooth systems of differential equations on an m-dimensional torus by A. M. Samoilenko [2] and later, independently, by J. Moser [3]. The Nash smoothing method, which was used in [2] (as indicated in [3-5]), leads to a large loss of smoothness. In the second part of the present paper, we use the Moser method of approximation of smooth functions by analytic ones for the reduction of smooth differential-difference equations on an m-dimensional torus to the canonical form
$$\dot \phi (t) = \omega + f(\phi (t),\;\phi (t - h)),{\text{}}f(\phi ,\;\phi - \omega h) = 0.$$
Note that the considered problem of investigation of differential-difference equations on an m-dimensional torus is important in theory [6, 7]. Moreover, the results obtained can be used for in the investigation of multifrequency oscillations of retarded systems.
Torus quasiperiodic solutions differential-difference equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  1. 1.Institute of Mathematics, National Academy of Sciences of UkraineKiev-4Ukraine

Personalised recommendations