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On the structure of the global attractor for non-autonomous dynamical systems with weak convergence

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  • The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so--called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert's question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations.
    Mathematics Subject Classification: Primary: 34C11, 34C27, 34D05, 34D23, 34D45, 34K14,37B20, 37B55, 37C55, 7C60, 37C65, 37C70, 37C75.


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