Quasiperiodic solutions of differential-difference equations on a torus
This interesting paper is dedicated to Victor A. Pliss on the occasion of his 70th birthday, who has proved a famous result concerning the structural stability of differential equations on a torus. This result has a closed connection with the problem considered here. In 1961, V. I. Arnold studied the reducibility of analytic systems on an \(m\)-dimensional torus to rotation. Then, it was extended to smooth systems of differential equations on an \(m\)-dimensional torus by A. M. Samoilenko in 1964 and later (in 1967), independently, by J. Moser. The present paper explores the quasi-periodic solutions of differential-difference equations on an \(m\)-dimensional torus and consists of two parts. The first part is devoted to extending V. I. Arnold's result to differential-difference equations on an \(m\)-dimensional torus. In the second part, the authors apply the Moser method of approximation of smooth functions by analytic ones to reduce the smooth differential-difference equations on an \(m\)-dimensional torus to the canonical form \[ \dot{\varphi}(t)=\omega+f(\varphi(t), \varphi(t-h)),\quad f(\varphi,\varphi-\omega h)=0. \] The results obtained here can be used to investigate the multifrequency oscillations of retarded systems.
scientific article; zbMATH DE number 2057853