Invariant Sets of Impulsive Differential Equations with Particularities in ω‐Limit Set

Abstract

Sufficient conditions for the existence and asymptotic stability of the invariant sets of an impulsive system of differential equations defined in the direct product of a torus and an Euclidean space are obtained.

On preservation of an exponentially stable invariant torus

“…Utilizing the Gronwall-Bellman type inequalities for different classes of functions it is easy to prove the analogous results for other classes of differential equations, in particular for impulsive equations (see [3] for details). Thus in [2] the analogue of Corollary 1 for a linear extension of dynamical system on torus with impulsive perturbations at non-fixed moments of time is proven.…”

Existence and Uniqueness of Mild Solutions for Nonlinear Stochastic Impulsive Differential Equation

“…Assume that p t 1 t t 3 and ψ x x, then it can be easily verified that the conditions H 1 -H 3 are satisfied, and by, Theorem 3.1, 4.1 has at least one solution.…”

Stability of trivial torus of nonlinear multifrequency systems via sign-definite quadratic forms

“…In this paper, we establish new conditions for exponential stability and instability of the trivial invariant torus of nonlinear extension of dynamical system on torus which are formulated in terms of quadratic forms that are sign-definite not on the entire surface of the torus, but in nonwandering set [7] of dynamical system on torus only. The corresponding results for linear extensions of dynamical systems on torus have been obtained in [1,3,[8][9][10][11]].…”