Perturbation Theorems for a Multifrequency System with Pulses

Feketa, Petro; Perestyuk, Yu. M.

doi:10.1007/s10958-016-2988-6

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…This paper is devoted to the study of an important class of evolutionary systems characterized by the presence of impulsive disturbances when the system trajectory reaches a predefined subset in the phase space, often termed impulsive dynamical systems [28,29]. The systematic study of these systems began relatively recently and primarily focused on systems defined in the finite-dimensional phase spaces, e.g., on systems defined in Euclidean space R n , n ∈ N [30][31][32][33][34] and the so-called multi-frequency systems defined in the product of a torus and Euclidean space T m × R n , n, m ∈ N [35][36][37][38]. The results regarding the limit behavior of infinite-dimensional impulsive dynamical systems can be found in [39][40][41][42][43][44][45].…”

Section: Introductionmentioning

confidence: 99%

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

Feketa,

Fedorenko,

Bezushchak

et al. 2023

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In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.

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Section: Introductionmentioning

confidence: 99%

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

Feketa,

Fedorenko,

Bezushchak

et al. 2023

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“…Recently, this framework was used due to application in biology and medicine [19]. Qualitative analysis of such systems in finite dimensional case was carried out in [24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Attractors for Multivalued Impulsive Systems: Existence and Applications to Reaction-Diffusion System

Капустян

Kapustyan

Gorban

2021

Mathematical Problems in Engineering

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In this paper, we develop a general approach to investigate limit dynamics of infinite-dimensional dissipative impulsive systems whose initial conditions do not uniquely determine their long time behavior. Based on the notion of an uniform attractor, we show how to describe limit behavior of such complex systems with the help of properties of their components. More precisely, we prove the existence of the uniform attractor for an impulsive multivalued system in terms of properties of nonimpulsive semiflow and impulsive parameters. We also give an application of these abstract results to the impulsive reaction-diffusion system without uniqueness.

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“…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]].…”

Section: Introductionmentioning

confidence: 99%