# American Institute of Mathematical Sciences

March  2012, 11(2): 809-828. doi: 10.3934/cpaa.2012.11.809

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

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received  January 2011 Revised  January 2011 Published  October 2011

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.
Citation: Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
##### References:
 [1] N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems,", Lecture Notes in Mathematics, (1970). Google Scholar [2] I. U. Bronsteyn, "Extensions of Minimal Transformation Group,", Noordhoff, (1979). Google Scholar [3] B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii, "Lyapunov Exponents Theory and Its Applications to Problems of Stabity,", Moscow, (1966). Google Scholar [4] T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations,, J. Differ. Eqns., 251 (2011). Google Scholar [5] D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients,, Differential Equations, 22 (1986), 267. Google Scholar [6] D. N. Cheban, $\mathbb C$-analytic dissipative dynamical systems,, Differential Equations, 22 (1986), 1915. Google Scholar [7] D. N. Cheban, Boundedness, dissipativity and almost periodicity of the solutions of linear and weakly nonlinear systems of differential equations,, Dynamical systems and boundary value problems, (1987), 143. Google Scholar [8] D. N. Cheban, Global pullback atttactors of C-analytic nonautonomous dynamical systems,, Stochastics and Dynamics, 1 (2001), 511. Google Scholar [9] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1., River Edge, (2004). Google Scholar [10] D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations,, J. Dynamics and Differential Equations, 20 (2008), 669. Google Scholar [11] D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations,", Hindawi Publishing Corporation, (2009). Google Scholar [12] D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems,", Nova Science Publishers, (2010). Google Scholar [13] D. N. Cheban and C. Mammana, Invariant manifolds, global attractors and almost periodic solutions of non-autonomous difference equations,, Nonlinear Analysis TMA, 56 (2004), 465. Google Scholar [14] D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations,, J. Math. Anal. Appl., 340 (2008), 374. Google Scholar [15] C. Conley, "Isolated Invariant Sets and the Morse Index,", Region. Conf. Ser. Math., (1978). Google Scholar [16] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I,, Vestnik MGU, 6 (1961), 19. Google Scholar [17] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II,, Vestnik MGU, 1 (1962), 3. Google Scholar [18] B. P. Demidovich, "Lectures on Mathematical Theory of Stability,", Moscow, (1967). Google Scholar [19] A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity,, Journal of Differential Equations, 9 (1971), 280. Google Scholar [20] J. K. Hale, "Asymptotic Behaviour of Dissipative Systems,", Amer. Math. Soc., (1988). Google Scholar [21] M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems,, J. Dyn. Diff. Eqns, 13 (2001), 107. Google Scholar [22] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge Univ. Press, (1982). Google Scholar [23] A. Pavlov, A. Pogrowsky, N. van de Wouw and N. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich,, Systems and Control Letters, 52 (2007), 257. Google Scholar [24] V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations,", Nauka, (1964). Google Scholar [25] V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations,", Nauka, (1977). Google Scholar [26] G. R. Sell, "Topological Dynamics and Ordinary Differential Equations,", Van Nostrand-Reinhold, (1971). Google Scholar [27] B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence,, Differential Equations, 11 (1975), 1246. Google Scholar [28] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations,", \cStiin\cta, (1985). Google Scholar [29] R. E. Vinograd, Inapplicability of the method of characteristic exponents to the study of non-linear differential equations,, Mat. Sb. N.S., 41 (1957), 431. Google Scholar [30] T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14,, Springer-Verlag, (1975). Google Scholar [31] V. V. Zhikov, On stability and unstability of Levinson's centre,, Differentsial'nye Uravneniya, 8 (1972), 2167. Google Scholar [32] V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations,, Mat. Sbornik, 90 (1973), 214. Google Scholar [33] V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application,", Noordhoof, (1964). Google Scholar [34] V. I. Zubov, "Theory of Oscillations,", Nauka, (1979). Google Scholar

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##### References:
 [1] N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems,", Lecture Notes in Mathematics, (1970). Google Scholar [2] I. U. Bronsteyn, "Extensions of Minimal Transformation Group,", Noordhoff, (1979). Google Scholar [3] B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii, "Lyapunov Exponents Theory and Its Applications to Problems of Stabity,", Moscow, (1966). Google Scholar [4] T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations,, J. Differ. Eqns., 251 (2011). Google Scholar [5] D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients,, Differential Equations, 22 (1986), 267. Google Scholar [6] D. N. Cheban, $\mathbb C$-analytic dissipative dynamical systems,, Differential Equations, 22 (1986), 1915. Google Scholar [7] D. N. Cheban, Boundedness, dissipativity and almost periodicity of the solutions of linear and weakly nonlinear systems of differential equations,, Dynamical systems and boundary value problems, (1987), 143. Google Scholar [8] D. N. Cheban, Global pullback atttactors of C-analytic nonautonomous dynamical systems,, Stochastics and Dynamics, 1 (2001), 511. Google Scholar [9] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1., River Edge, (2004). Google Scholar [10] D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations,, J. Dynamics and Differential Equations, 20 (2008), 669. Google Scholar [11] D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations,", Hindawi Publishing Corporation, (2009). Google Scholar [12] D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems,", Nova Science Publishers, (2010). Google Scholar [13] D. N. Cheban and C. Mammana, Invariant manifolds, global attractors and almost periodic solutions of non-autonomous difference equations,, Nonlinear Analysis TMA, 56 (2004), 465. Google Scholar [14] D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations,, J. Math. Anal. Appl., 340 (2008), 374. Google Scholar [15] C. Conley, "Isolated Invariant Sets and the Morse Index,", Region. Conf. Ser. Math., (1978). Google Scholar [16] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I,, Vestnik MGU, 6 (1961), 19. Google Scholar [17] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II,, Vestnik MGU, 1 (1962), 3. Google Scholar [18] B. P. Demidovich, "Lectures on Mathematical Theory of Stability,", Moscow, (1967). Google Scholar [19] A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity,, Journal of Differential Equations, 9 (1971), 280. Google Scholar [20] J. K. Hale, "Asymptotic Behaviour of Dissipative Systems,", Amer. Math. Soc., (1988). Google Scholar [21] M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems,, J. Dyn. Diff. Eqns, 13 (2001), 107. Google Scholar [22] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge Univ. Press, (1982). Google Scholar [23] A. Pavlov, A. Pogrowsky, N. van de Wouw and N. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich,, Systems and Control Letters, 52 (2007), 257. Google Scholar [24] V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations,", Nauka, (1964). Google Scholar [25] V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations,", Nauka, (1977). Google Scholar [26] G. R. Sell, "Topological Dynamics and Ordinary Differential Equations,", Van Nostrand-Reinhold, (1971). Google Scholar [27] B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence,, Differential Equations, 11 (1975), 1246. Google Scholar [28] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations,", \cStiin\cta, (1985). Google Scholar [29] R. E. Vinograd, Inapplicability of the method of characteristic exponents to the study of non-linear differential equations,, Mat. Sb. N.S., 41 (1957), 431. Google Scholar [30] T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14,, Springer-Verlag, (1975). Google Scholar [31] V. V. Zhikov, On stability and unstability of Levinson's centre,, Differentsial'nye Uravneniya, 8 (1972), 2167. Google Scholar [32] V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations,, Mat. Sbornik, 90 (1973), 214. Google Scholar [33] V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application,", Noordhoof, (1964). Google Scholar [34] V. I. Zubov, "Theory of Oscillations,", Nauka, (1979). Google Scholar
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