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

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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

Tomás Caraballo ^1, and David Cheban ^2,

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

Abstract
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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.

Keywords: dissipative systems, convergent systems, skew-product systems, almost periodic, global attractor, Non-autonomous dynamical systems, cocycles, almost automorphic, quasi-periodic, asymptotically almost periodic solutions., recurrent solutions.

Mathematics Subject Classification: Primary: 34C11, 34C27, 34D05, 34D23, 34D45, 34K14,37B20, 37B55, 37C55, 7C60, 37C65, 37C70, 37C7.

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).
[2]	I. U. Bronsteyn, "Extensions of Minimal Transformation Group,", Noordhoff, (1979).
[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).
[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).
[5]	D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients,, Differential Equations, 22 (1986), 267.
[6]	D. N. Cheban, $\mathbb C$-analytic dissipative dynamical systems,, Differential Equations, 22 (1986), 1915.
[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.
[8]	D. N. Cheban, Global pullback atttactors of C-analytic nonautonomous dynamical systems,, Stochastics and Dynamics, 1 (2001), 511.
[9]	D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1., River Edge, (2004).
[10]	D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations,, J. Dynamics and Differential Equations, 20 (2008), 669.
[11]	D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations,", Hindawi Publishing Corporation, (2009).
[12]	D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems,", Nova Science Publishers, (2010).
[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.
[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.
[15]	C. Conley, "Isolated Invariant Sets and the Morse Index,", Region. Conf. Ser. Math., (1978).
[16]	B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I,, Vestnik MGU, 6 (1961), 19.
[17]	B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II,, Vestnik MGU, 1 (1962), 3.
[18]	B. P. Demidovich, "Lectures on Mathematical Theory of Stability,", Moscow, (1967).
[19]	A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity,, Journal of Differential Equations, 9 (1971), 280.
[20]	J. K. Hale, "Asymptotic Behaviour of Dissipative Systems,", Amer. Math. Soc., (1988).
[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.
[22]	B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge Univ. Press, (1982).
[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.
[24]	V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations,", Nauka, (1964).
[25]	V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations,", Nauka, (1977).
[26]	G. R. Sell, "Topological Dynamics and Ordinary Differential Equations,", Van Nostrand-Reinhold, (1971).
[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.
[28]	B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations,", \cStiin\cta, (1985).
[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.
[30]	T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14,, Springer-Verlag, (1975).
[31]	V. V. Zhikov, On stability and unstability of Levinson's centre,, Differentsial'nye Uravneniya, 8 (1972), 2167.
[32]	V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations,, Mat. Sbornik, 90 (1973), 214.
[33]	V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application,", Noordhoof, (1964).
[34]	V. I. Zubov, "Theory of Oscillations,", Nauka, (1979).

show all references

References:

[1]	N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems,", Lecture Notes in Mathematics, (1970).
[2]	I. U. Bronsteyn, "Extensions of Minimal Transformation Group,", Noordhoff, (1979).
[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).
[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).
[5]	D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients,, Differential Equations, 22 (1986), 267.
[6]	D. N. Cheban, $\mathbb C$-analytic dissipative dynamical systems,, Differential Equations, 22 (1986), 1915.
[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.
[8]	D. N. Cheban, Global pullback atttactors of C-analytic nonautonomous dynamical systems,, Stochastics and Dynamics, 1 (2001), 511.
[9]	D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1., River Edge, (2004).
[10]	D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations,, J. Dynamics and Differential Equations, 20 (2008), 669.
[11]	D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations,", Hindawi Publishing Corporation, (2009).
[12]	D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems,", Nova Science Publishers, (2010).
[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.
[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.
[15]	C. Conley, "Isolated Invariant Sets and the Morse Index,", Region. Conf. Ser. Math., (1978).
[16]	B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I,, Vestnik MGU, 6 (1961), 19.
[17]	B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II,, Vestnik MGU, 1 (1962), 3.
[18]	B. P. Demidovich, "Lectures on Mathematical Theory of Stability,", Moscow, (1967).
[19]	A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity,, Journal of Differential Equations, 9 (1971), 280.
[20]	J. K. Hale, "Asymptotic Behaviour of Dissipative Systems,", Amer. Math. Soc., (1988).
[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.
[22]	B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge Univ. Press, (1982).
[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.
[24]	V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations,", Nauka, (1964).
[25]	V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations,", Nauka, (1977).
[26]	G. R. Sell, "Topological Dynamics and Ordinary Differential Equations,", Van Nostrand-Reinhold, (1971).
[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.
[28]	B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations,", \cStiin\cta, (1985).
[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.
[30]	T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14,, Springer-Verlag, (1975).
[31]	V. V. Zhikov, On stability and unstability of Levinson's centre,, Differentsial'nye Uravneniya, 8 (1972), 2167.
[32]	V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations,, Mat. Sbornik, 90 (1973), 214.
[33]	V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application,", Noordhoof, (1964).
[34]	V. I. Zubov, "Theory of Oscillations,", Nauka, (1979).

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