On the existence of invariant tori of countable systems of difference-differential equations defined on infinite-dimensional tori

Abstract

UDC 517.9In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for linear and quasilinear countable systems of differential-difference equations defined on infinitedimensional tori and containing an infinite set of constant deviations of a scalar argument. Statement of the ProblemIt is known that the investigation of invariant sets (in particular, invariant tori) occupies an important place both in the theory of continuous dynamical systems (flows) and in the theory of discrete dynamical systems (cascades) defined in various normed spaces. In the last four decades, numerous fundamental results were obtained in this field of mathematics with the use of the method of the Green function of the problem of an invariant torus of a linear expansion of a dynamical system proposed by Samoilenko in 1970 (see [1,2]). In [3-6], this method was used for the investigation of invariant tori of countable systems of ordinary differential equations defined on tori. In the last ten years, several works were published (see [7][8][9][10][11][12][13][14][15]) in which this method was used for the investigation of invariant tori of countable systems of difference-differential and difference equations. In the present paper, in the space of bounded number sequences, we pose and solve the problem of finding sufficient conditions for the existence of invariant tori for linear and quasilinear countable systems of difference-differential equations defined on infinite-dimensional tori and containing an infinite set of constant different-sign deviations of a scalar argument. This problem has never been investigated before in the mathematical literature.

On the existence of infinite-dimensional invariant tori of nonlinear countable systems of difference-differential equations

“…The theorems proved in this paper agree well with the results of [1,2] concerning the existence of invariant tori of countable systems of differential and difference equations defined on tori. This paper continues the investigations carried out in [3][4][5], where analogous problems were solved for linear and quasilinear systems of the indicated type. .…”

“…; / and G 0 . ; / for the problem of bounded solutions and the problem of invariant tori, respectively, was considered in detail in [1,3] for a countable system of equations of the form…”