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Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations. (English) Zbl 1065.39026

The authors study quasilinear nonautonomous difference equations. They prove that such equations admit an invariant manifold. Further, conditions guaranteeing the existence of a compact global attractor are obtained. Its structure is characterized. A criterion for the existence of almost periodic and recurrent solutions of the quasilinear nonautonomous difference equations is also derived. Finally, it is proved that quasilinear maps with chaotic basis admit a chaotic compact invariant set.
Reviewer: Pavel Rehak (Brno)

MSC:

39A12 Discrete version of topics in analysis
37D10 Invariant manifold theory for dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Blanchard, F.; Glasner, E.; Kolyada, S.; Mass, A., On Li-Yorke pairs, J. Reine Angew. Mat. (Crelle’s Journal), 547, 51-68 (2002) · Zbl 1059.37006
[2] Cheban, D. N., Global Attractors of Nonautonomous Dissipative Dynamical Systems (2003), World Scientific: World Scientific Singapore, to appear · Zbl 1024.37010
[3] Cheban, D. N., Global Attractors of Nonautonomous Dynamical Systems (2002), State University of Moldova: State University of Moldova Moldova, (in Russian) · Zbl 1024.37010
[4] Cheban, D. N.; Duan, J.; Gherco, A., Generalization of second Bogoliubov’s theorem for non-almost periodic systems, Nonlinear Anal. Real World Appl., 4, 4, 599-613 (2003) · Zbl 1027.34065
[5] C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamicals Systems and Differential Equations, American Mathematical Society, Providence, RI, 1999.; C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamicals Systems and Differential Equations, American Mathematical Society, Providence, RI, 1999. · Zbl 0970.47027
[6] I.D. Chueshov, Introduction into the theory of infinite-dimensional dissipative systems, Acta, Kharkiv, 2002.; I.D. Chueshov, Introduction into the theory of infinite-dimensional dissipative systems, Acta, Kharkiv, 2002. · Zbl 1100.37047
[7] Glendinning, P., Global attractors of pinched skew-products, Dynamical Systems, 17, 23, 287-294 (2002) · Zbl 1024.37017
[8] A. Halanay, D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Bucureşti, 1968.; A. Halanay, D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Bucureşti, 1968. · Zbl 0176.05202
[9] Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840 (1981), Springer: Springer Berlin · Zbl 0456.35001
[10] Kloeden, P. E., On Sharkovsky’s cycle coexistence ordering, Bull. Austral. Math. Soc., 20, 171-177 (1979) · Zbl 0465.58022
[11] Kolyada, S., On dynamics of triangular maps of square, Ergodic Theory Dynamical Systems, 12, 749-768 (1992) · Zbl 0784.58038
[12] Robinson, C., Dynamical Systems, Stability, Symbolic Dynamics and Chaos (1995), CRC Press: CRC Press Boca Raton Ann Arbor, London, Tokyo · Zbl 0853.58001
[13] Sell, G. R., Topological Dynamics and Ordinary Differential Equations (1971), Van Nostrand, Reinhold: Van Nostrand, Reinhold London · Zbl 0212.29202
[14] Sharkovsky, A. N.; Maistrenko, Yu. L.; Romanenko, E. Yu., Difference Equations and Their Applications (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0806.39001
[15] Shcherbakov, B. A., Topological Dynamics and Poisson’s Stability of Solutions of Differential Equations (1972), Kishinev: Kishinev Shtiintsa, (in Russian) · Zbl 0256.34062
[16] Shcherbakov, B. A., Poisson’s Stability of Motions of Dynamical Systems and Solutions of Differential Equations (1985), Kishinev: Kishinev Shtiintsa, (in Russian) · Zbl 0638.34046
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