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Shao, Hua; Chen, Guanrong; Shi, Yuming

Some criteria of chaos in non-autonomous discrete dynamical systems. (English) Zbl 1440.37028

J. Difference Equ. Appl. 26, No. 3, 295-308 (2020).
This paper is devoted to the study of chaos in nonautonomous discrete systems of the following kind: \[ x_{n+1}=f_{n}(x_{n})\ \qquad n\in \mathbb Z_{+}, \tag{1} \] where \(f_n\) is a map from \(X\) to \(X\) for any \(n\in \mathbb Z_{+}\) and \(X\) is a metric space.
For the system (1) the authors study the following types of chaotic behavior: Li-Yorke \(\delta\)-chaos, strong Li-Yorke chaos and distributional \(\delta\)-chaos.
Reviewer: David Cheban (Chisinau)

Cited in 1 Document

MSC:

37B55 Topological dynamics of nonautonomous systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B10 Symbolic dynamics
39A33 Chaotic behavior of solutions of difference equations

Keywords:

nonautonomous discrete dynamical system; strong Li-Yorke chaos; distributional chaos; coupled-expansion; irreducible transition matrix
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References:

[1] Cánovas, J. S., Li-Yorke chaos in a class of non-autonomous discrete systems, J. Differ. Equ. Appl., 17, 479-486 (2011) · Zbl 1221.37080 · doi:10.1080/10236190903049025
[2] Huang, Q.; Shi, Y.; Zhang, L., Sensitivity of non-autonomous discrete dynamical systems, Appl. Math. Lett., 39, 31-34 (2015) · Zbl 1370.37031 · doi:10.1016/j.aml.2014.08.007
[3] Ju, H.; Shao, H.; Choe, Y.; Shi, Y., Conditions for maps to be topologically conjugate or semi-conjugate to subshifts of finite type and criteria of chaos, Dyn. Syst., 31, 496-505 (2016) · Zbl 1367.37021 · doi:10.1080/14689367.2016.1158240
[4] Ju, H.; Kim, C.; Choe, Y.; Chen, M., Conditions for topologically semi-conjugacy of the induced systems to the subshift of finite type, Chaos Solitons Fractals, 98, 1-6 (2017) · Zbl 1372.37021 · doi:10.1016/j.chaos.2017.02.012
[5] Kennedy, J.; Yorke, J. A., Topological horseshoes, Trans. Am. Math. Soc., 353, 2513-2531 (2001) · Zbl 0972.37011 · doi:10.1090/S0002-9947-01-02586-7
[6] Kim, C.; Ju, H.; Chen, M.; Raith, P., A-Coupled-expanding and distributional chaos, Chaos Solitons Fractals, 77, 291-295 (2015) · Zbl 1353.37028 · doi:10.1016/j.chaos.2015.06.010
[7] Kolyada, S.; Snoha, L., Topological entropy of non-autonomous dynamical systems, Random Comput. Dyn., 4, 205-233 (1996) · Zbl 0909.54012
[8] Kulczycki, M.; Oprocha, P., Coupled-expanding maps and matrix shifts, Internat. J. Bifur. Chaos, 23, 1-6 (2013) · Zbl 1270.37015 · doi:10.1142/S0218127413500132
[9] Robinson, C., Dynamical Systems: Stability Symbolic Dynamics and Chaos, 2nd ed. (1999), CRC Press: CRC Press, Florida · Zbl 0914.58021
[10] Shao, H.; Shi, Y.; Zhu, H., Strong Li-Yorke chaos for time-varying discrete systems with A-coupled-expansion, Internat. J. Bifur. Chaos, 25 (2015) · Zbl 1330.37036 · doi:10.1142/S0218127415501862
[11] Shao, H.; Shi, Y.; Zhu, H., Lyapunov exponents, sensitivity, and stability for non-autonomous discrete systems, Internat. J. Bifur. Chaos, 28 (2018) · Zbl 1392.37025 · doi:10.1142/S0218127418500888
[12] Shao, H.; Shi, Y.; Zhu, H., On distributional chaos in non-autonomous discrete systems, Chaos Solitons Fractals, 107, 234-243 (2018) · Zbl 1380.37038 · doi:10.1016/j.chaos.2018.01.005
[13] Shi, Y., Chaos in non-autonomous discrete dynamical systems approached by their induced systems, Internat. J. Bifur. Chaos, 22 (2012) · Zbl 1258.37023
[14] Shi, Y.; Chen, G., Chaos of discrete dynamical systems in complete metric spaces, Chaos Solitons Fractals, 22, 555-571 (2004) · Zbl 1067.37047 · doi:10.1016/j.chaos.2004.02.015
[15] Shi, Y.; Chen, G., Chaos of time-varying discrete dynamical systems, J. Differ. Equ. Appl., 15, 429-449 (2009) · Zbl 1173.37032 · doi:10.1080/10236190802020879
[16] Shi, Y.; Ju, H.; Chen, G., Coupled-expanding maps and one-sided symbolic dynamical systems, Chaos Solitons Fractals, 39, 2138-2149 (2009) · Zbl 1197.37010 · doi:10.1016/j.chaos.2007.06.090
[17] Tian, C.; Chen, G., Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals, 28, 1067-1075 (2006) · Zbl 1095.54018 · doi:10.1016/j.chaos.2005.08.127
[18] Wang, L.; Liao, G.; Yang, Y., Recurrent point set of the shift on Σ and strong chaos, Ann. Polon. Math., LXX VIII 2, 124-130 (2002)
[19] Wu, X.; Zhu, P., Chaos in a class of non-autonomous discrete systems, Appl. Math. Lett., 26, 431-436 (2013) · Zbl 1261.39012 · doi:10.1016/j.aml.2012.11.003
[20] Yang, X.; Tang, Y., Horseshoes in piecewise continuous maps, Chaos Solitons Fractals, 19, 841-845 (2004) · Zbl 1053.37006 · doi:10.1016/S0960-0779(03)00202-9
[21] Zhang, X.; Shi, Y., Coupled-expanding maps for irreducible transition matrices, Int. J. Bifurcat. Chaos, 20, 3769-3783 (2010) · Zbl 1208.37012 · doi:10.1142/S0218127410028094
[22] Zhang, X.; Shi, Y.; Chen, G., Some properties of coupled-expanding maps in compact sets, Proc. Am. Math. Soc., 141, 585-595 (2013) · Zbl 1353.37021 · doi:10.1090/S0002-9939-2012-11339-5
[23] Zhang, L.; Shi, Y.; Shao, H.; Huang, Q., Chaos induced by weak A-coupled-expansion of non-autonomous discrete dynamical systems, J. Differ. Equ. Appl., 22, 1747-1759 (2016) · Zbl 1375.37053 · doi:10.1080/10236198.2016.1243104
[24] Zhou, Z., Symbolic Dynamics (1997), Shanghai Scientific and Technological Education Publishing House: Shanghai Scientific and Technological Education Publishing House, Shanghai
[25] Zhu, Y.; Liu, Z.; Xu, X.; Zhang, W., Entropy of non-autonomous dynamical systems, J. Korean Math. Soc., 49, 165-185 (2012) · Zbl 1252.37009 · doi:10.4134/JKMS.2012.49.1.165
[26] Zhu, H.; Shi, Y.; Shao, H., Devaney chaos in nonautonomous discrete systems, Internat. J. Bifur. Chaos, 26 (2016) · Zbl 1349.37044
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