Facultatea de Matematică şi Informatică / Faculty of Methematics and Informatics
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Item INVARIANT MANIFOLDS, GLOBAL ATTRACTORS AND ALMOST PERIODIC SOLUTIONS OF NONAUTONOMOUS DIFFERENCE EQUATIONS(Elsevier, 2004) Cheban, David; Mammana, CristianaThe 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.Item ALMOST PERIODIC SOLUTIONS AND GLOBAL ATTRACTORS OF NON-AUTONOMOUS NAVIER–STOKES EQUATIONS(Springer, 2004) Cheban, David; Duan, JinqiaoThe 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.Item GLOBAL ATTRACTORS FOR V -MONOTONE NONAUTONOMOUS DYNAMICAL SYSTEMS(Institutul de Matematică şi Informatică al AŞM, 2003) Cheban, David; Kloeden, Peter-E.; Schmalfuss, BjornThis 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.Item ASYMPTOTIC STABILITY OF AUTONOMOUSAND NON-AUTONOMOUS DISCRETE LINEAR INCLUSIONS(Institutul de Matematică şi Informatică al AŞM, 2004) Cheban, David; Mammana, CristianaThe 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).Item ABSOLUTE ASYMPTOTIC STABILITY OF DISCRETE LINEAR INCLUSIONS(Institutul de Matematică şi Informatică al AŞM, 2006) Cheban, David; Mammana, CristianaThe 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).Item GLOBAL ATTRACTORS OF NON-AUTONOMOUS DIFFERENCE EQUATIONS(Institutul de Matematică şi Informatică al AŞM, 2009) Cheban, David; Mammana, Cristina; Michetti, ElizabettaThe 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