Browsing by Author "Cheban, David"

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ABSOLUTE ASYMPTOTIC STABILITY OF DISCRETE LINEAR INCLUSIONS
(Institutul de Matematică şi Informatică al AŞM, 2006) Cheban, David; Mammana, Cristiana
The article is devoted to the study of absolute asymptotic stability of discrete linear inclusions in Banach (both finite and infinite dimensional) space. We establish the relation between absolute asymptotic stability, asymptotic stability, uniform asymptotic stability and uniform exponential stability. It is proved that for asymptotical compact (a sum of compact operator and contraction) discrete linear inclusions the notions of asymptotic stability and uniform exponential stability are equivalent. It is proved that finite-dimensional discrete linear inclusion, defined by matrices {A1,A2, ...,Am}, is absolutely asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the matrices {A1,A2, ...,Am} is asymptotically stable. We study this problem in the framework of non-autonomous dynamical systems (cocyles).
ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF MONOTONE DIFFERENTIAL EQUATIONS WITH A STRICT MONOTONE FIRST INTEGRAL
(Institutul de Matematică şi Informatică al AŞM, 2020) Cheban, David
The paper is dedicated to the study of problem of Poisson stability (in particular periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, Levitan almost periodicity, pseudo-periodicity, almost recurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, pseudo-recurrence, Poisson stability) and asymptotical Poisson stability of motions of monotone non-autonomous differential equations which admit a strict monotone first integral. This problem is solved in the framework of general non-autonomous dynamical systems.
ALMOST PERIODIC SOLUTIONS AND GLOBAL ATTRACTORS OF NON-AUTONOMOUS NAVIER–STOKES EQUATIONS
(Springer, 2004) Cheban, David; Duan, Jinqiao
The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.
ASYMPTOTIC STABILITY OF AUTONOMOUSAND NON-AUTONOMOUS DISCRETE LINEAR INCLUSIONS
(Institutul de Matematică şi Informatică al AŞM, 2004) Cheban, David; Mammana, Cristiana
The article is devoted to the study of absolute asymptotic stability ofdiscrete linear inclusions (both autonomous and non-autonomous) in Banach space.We establish the relation between absolute asymptotic stability, uniform asymptoticstability and uniform exponential stability. It is proved that for compact (completelycontinuous) discrete linear inclusions these notions of stability are equivalent. Westudy this problem in the framework of non-autonomous dynamical systems (cocyles).
AVERAGING PRINCIPLE ON INFINITE INTERVALS FOR STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS
(AIMS Press, 2021) Cheban, David; Zhenxin, Liu
In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.
AVERAGING PRINCIPLE ON SEMI-AXIS FOR SEMI-LINEAR DIFFERENTIAL EQUATIONS
(Casa Editorial-Poligrafică „Bons Offices”, 2022) Cheban, David
BIRKHOFF'S CENTER OF COMPACT DISSIPATIVE DYNAMICAL SYSTEMS
(Valines SRL, 2014) Cheban, David
We introduce the notion of Birkhoff center for arbitrary dynamical systems admitting a compact global attractor. It is shown that Birkhoff center of dynamical system coincides with the closure of the set of all positively Poisson stable points of dynamical system.
GLOBAL ATTRACTORS FOR V -MONOTONE NONAUTONOMOUS DYNAMICAL SYSTEMS
(Institutul de Matematică şi Informatică al AŞM, 2003) Cheban, David; Kloeden, Peter-E.; Schmalfuss, Bjorn
This article is devoted to the study of the compact global atrractors of Vmomotone nonautonomous dynamical systems.We give a description of the structure of compact global attractors of this class of systems. Several applications of general results for different classes of differential equations (ODEs, ODEs with impulse, some classes of evolutionary partial differential equations) are given.
GLOBAL ATTRACTORS OF NON-AUTONOMOUS DIFFERENCE EQUATIONS
(Institutul de Matematică şi Informatică al AŞM, 2009) Cheban, David; Mammana, Cristina; Michetti, Elizabetta
The article is devoted to the study of global attractors of quasi-linear non-autonomous di®erence equations. The results obtained are applied to the study of a triangular economic growth model T : R2 ! R2 recently developed in S. Brianzoni, C. Mammana and E. Michetti [1
INVARIANT MANIFOLDS, GLOBAL ATTRACTORS AND ALMOST PERIODIC SOLUTIONS OF NONAUTONOMOUS DIFFERENCE EQUATIONS
(Elsevier, 2004) Cheban, David; Mammana, Cristiana
The article is devoted to the study of quasi-linear nonautonomous difference equations: invariant manifolds, compact global attractors, almost periodic and recurrent solutions and chaotic sets. First, we prove that such equations admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and recurrent solutions of the quasi-linear nonautonomous difference equations. Finally, we prove that quasi-linear maps with chaotic base admit a chaotic compact invariant set. The obtained results are applied while studying triangular maps: invariant manifolds, compact global attractors, almost periodic and recurrent solutions and chaotic sets.
LEVITAN ALMOST PERIODIC SOLUTIONS OF INFINITE-DIMENSIONAL LINEAR DIFFERENTIAL EQUATIONS
(Institutul de Matematică şi Informatică al AŞM, 2019) Cheban, David
The known Levitan’s Theorem states that the finite-dimensional linear differential equationx′=A(x+f(t)(1)with Bohr almost periodic coefficientsA(t) and f(t) admits at least one Levitan almostperiodic solution if it has a bounded solution. The main assu mption in this theoremis the separation among bounded solutions of homogeneous eq uationsx′=A(t)x .(2)In this paper we prove that infinite-dimensional linear differential equation (3) withLevitan almost periodic coefficients has a Levitan almost periodic solution if it has at least one relatively compact solution and the trivial solut ion of equation (2) is Lyapunov stable. We study the problem of existence of Bohr/Levi tan almost periodicsolutions for infinite-dimensional equation (3) in the fram ework of general nonau tonomous dynamical systems (cocycles).
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF SCALAR DIFFERENTIAL EQUATIONS
(Taylor & Francis, 2018) Cheban, David
The aim of this paper is to prove the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the scalar differential equations. The existence of at least one quasi-periodic (respectively, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent, Levitan almost periodic, almost recurrent, Poisson stable) solution of sclalar differential equations is proved under the condition that it admits at least one bounded solution on the positive semi-axis which is uniformly Lyapunov stable.
NON-AUTONOMOUS DYNAMICAL SYSTEMS AND THEIR APPLICATIONS
(Academia de Ştiinţe a Moldovei, 2021) Cheban, David
Articolul reprezintă o scurtă trecere în revistă a cercetărilor efectuate de autor în ultimii 10-15 ani privind sistemele dinamice neautonome şi aplicațiile acestora. Sistemele dinamice neautonome constituie un nou domeniu ce contribuie la dezvoltarea rapidă a matematicii (teoria sistemelor dinamice). Mii de articole, inclusiv zeci de articole de sinteză și un șir de monografii despre sistemele dinamice neautonome au fost publicate în ultimele decenii, iar problematica respectivă a făcut cap de afiș la conferințele internaționale. Autorul a publicat trei monografii pe problema sistemelor dinamice neautonome. În acest articol este oferită o prezentare generală a rezultatelor obținute.
POISSON STABLE MOTIONS OF MONOTONE AND STRONGLY SUB-LINEAR NON-AUTONOMOUS DYNAMICAL SYSTEMS
(Hybrid & Monthly, 2023) Cheban, David
This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo periodic, pseudo recurrent and Poisson stable) motions of monotone sub-linear non-autonomous dynamical systems. The main results we establish in the framework of general non-autonomous (cocycle) dynamical systems. We apply our general results to the study of the problem of existence of different classes Poisson stable solutions of some types of non-autonomous evolutionary equations (Ordinary Differential Equations, Functional-Differential Equations with finite delay and Difference Equations).
RELATION BETWEEN LEVINSON CENTER, CHAIN RECURRENT SET AND CENTER OF BIRKHOFF FOR COMPACT DISSIPATIVE DYNAMICAL SYSTEMS
(Academy of Sciences of Moldova, 2015) Cheban, David
In this paper we prove the analogues of Birkhoff’s theorem for one- sided dynamical systems (both with continuous and discrete times) with noncompact space having a compact global attractor. The relation between Levinson center, chain recurrent set and center of Birkhoff is established for compact dissipative dynamical systems. Mathematics subject classification: 37B25, 37B35 37B55, 37L15, 37L30, 37L45.