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Chaos of time-varying discrete dynamical systems. (English) Zbl 1173.37032

Authors’ abstract: This paper is concerned with chaos of time-varying (i.e. non-autonomous) discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including periodic point, coupled-expansion for transitive matrix, uniformly topological equiconjugacy, and three definitions of chaos, i.e. chaos in the sense of Devaney and Wiggins, respectively, and in a strong sense of Li-Yorke. An interesting observation is that a finite-dimensional linear time-varying system can be chaotic in the original sense of Li-Yorke, but cannot have chaos in the strong sense of Li-Yorke, nor in the sense of Devaney in a set containing infinitely many points, and nor in the sense of Wiggins in a set starting from which all the orbits are bounded. A criterion of chaos in the original sense of Li-Yorke is established for finite-dimensional linear time-varying systems. Some basic properties of topological conjugacy are discussed. In particular, it is shown that topological conjugacy alone cannot guarantee two topologically conjugate time-varying systems to have the same topological properties in general. In addition, a criterion of chaos induced by strict coupled-expansion for a certain irreducible transitive matrix is established, under which the corresponding nonlinear system is proved chaotic in the strong sense of Li-Yorke. Two illustrative examples are finally provided with computer simulations for illustration.
Reviewer: Tingwen Huang

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B55 Topological dynamics of nonautonomous systems
37B10 Symbolic dynamics
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[1] AlSharawi Z., J. Math. Anal. Appl.
[2] DOI: 10.3934/dcdsb.2005.5.215 · Zbl 1076.37011 · doi:10.3934/dcdsb.2005.5.215
[3] DOI: 10.2307/2324899 · Zbl 0758.58019 · doi:10.2307/2324899
[4] DOI: 10.1090/S0002-9939-02-06762-X · Zbl 1044.47006 · doi:10.1090/S0002-9939-02-06762-X
[5] Block L.S., Dynamics in One Dimension, Lecture Notes in Mathematics 1513 (1992) · Zbl 0746.58007 · doi:10.1007/BFb0084762
[6] DOI: 10.1017/S0143385797084976 · Zbl 0910.47033 · doi:10.1017/S0143385797084976
[7] Devaney R.L., An Introduction to Chaotic Dynamical Systems (1987)
[8] Dewilde P., Time-varying Systems and Computations (1998) · Zbl 0937.93002 · doi:10.1007/978-1-4757-2817-0
[9] DOI: 10.1016/j.jde.2003.10.024 · Zbl 1067.39003 · doi:10.1016/j.jde.2003.10.024
[10] DOI: 10.1016/S0166-8641(01)00025-6 · Zbl 0997.54061 · doi:10.1016/S0166-8641(01)00025-6
[11] DOI: 10.1109/TAC.2003.812781 · Zbl 1364.93514 · doi:10.1109/TAC.2003.812781
[12] DOI: 10.1090/S0002-9939-98-04344-5 · Zbl 0893.54033 · doi:10.1090/S0002-9939-98-04344-5
[13] DOI: 10.1080/10236190600574069 · Zbl 1096.39019 · doi:10.1080/10236190600574069
[14] DOI: 10.2307/2589144 · Zbl 0992.37029 · doi:10.2307/2589144
[15] DOI: 10.2307/2318254 · Zbl 0351.92021 · doi:10.2307/2318254
[16] DOI: 10.1016/0022-247X(78)90115-4 · Zbl 0381.58004 · doi:10.1016/0022-247X(78)90115-4
[17] DOI: 10.1016/j.chaos.2004.10.003 · Zbl 1077.37027 · doi:10.1016/j.chaos.2004.10.003
[18] DOI: 10.2307/2691012 · Zbl 1008.37014 · doi:10.2307/2691012
[19] DOI: 10.1109/TAC.2004.841888 · Zbl 1365.93268 · doi:10.1109/TAC.2004.841888
[20] Robinson C., Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2. ed. (1999) · Zbl 0914.58021
[21] DOI: 10.1016/j.chaos.2004.02.015 · Zbl 1067.37047 · doi:10.1016/j.chaos.2004.02.015
[22] DOI: 10.1360/03ys0183 · Zbl 1170.37306 · doi:10.1360/03ys0183
[23] ChenG., Coupled-expanding maps and one-sided symbolic dynamical systems, Chaos Solitons Fractals, accepted for publication
[24] DOI: 10.1016/j.chaos.2005.08.162 · Zbl 1106.37008 · doi:10.1016/j.chaos.2005.08.162
[25] Yu P., Dynamics of Continuous, Discrete, and Implusive Systems, Series B: Applications and Algorithms 14 pp 175– (2007)
[26] DOI: 10.1016/j.jmaa.2007.05.005 · Zbl 1131.37023 · doi:10.1016/j.jmaa.2007.05.005
[27] DOI: 10.1090/S0002-9904-1967-11798-1 · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[28] DOI: 10.1016/j.chaos.2005.08.127 · Zbl 1095.54018 · doi:10.1016/j.chaos.2005.08.127
[29] Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (1990) · Zbl 0701.58001 · doi:10.1007/978-1-4757-4067-7
[30] Zhou Z., Symbolic Dynamics (1997)
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