Stability of invariant manifold of nonlinear system of differential equations

Abstract

The theory of extensions of the dynamical equations on the torus is an important section of the theory of ordinary differential equations that is intensively evolving and has an important applied application to various tasks of science and technology. This theory describes processes that have oscillatory character. One of the important questions of mathematical theory of multifrequency oscillations is the problem of the existence and stability of invariant toroidal manifolds, the problem of the structural stability of the invariant manifold, its preservation under small perturbations for the systems of differential equations that are defined in the direct product of a torus and Euclidean space. Such manifolds serve as carriers of multifrequency oscillations in the system. The basics of this theory have been developed by A. M. Samoilenko. In this paper the class of differential equations, defined in the direct product of m-dimensional torus T m and n-dimensional Euclidean space R n for which there exist conditions for the existence of an asymptotically stable invariant toroidal manifold, is investigated. We formulate and prove sufficient conditions for the existence and asymptotic stability of invariant toroidal class of nonlinear extensions of dynamical systems on torus, that has specific properties in ω-limit set Ω of the trajectories ϕ_t (ϕ). Two theorems, that define the conditions for the existence of asymptotically stable invariant sets for linear expansion of the dynamical system on torus and corresponding perturbed system, are given. We search an invariant manifold of a nonlinear system by the iterative method, proposed by A. M. Samoilenko.

Keywords

• нелiнiйна система диференцiальних рiвнянь
• iнварiантний многовид
• асимптотична стiйкiсть
• m-вимiрний тор
• n-вимiрний евклiдiв простiр
• ω-гранична множина
• iтерацiйний метод