THE CHAIN PROPERTIES AND LI-YORKE SENSITIVITY OF ZADEH'S EXTENSION ON THE SPACE OF UPPER SEMI-CONTINUOUS FUZZY SETS

Document Type: Research Paper

Authors

1 School of Sciences, Southwest Petroleum University, Chengdu, Sichuan, 610500, People's Republic of China

2 Zhuhai College of Jilin University, Zhuhai, Guangdong, 519041, Peoples Republic of China

3 School of Mathematical Sciences, Dalian University of Technology, Liaoning, Dalian, 116024, People's Republic of China

Abstract

Some characterizations on the chain recurrence, chain transitivity, chain mixing property,
shadowing and $h$-shadowing for Zadeh's extension are obtained. Besides, it is proved
that a dynamical system is spatiotemporally chaotic provided that the Zadeh's extension
is Li-Yorke sensitive.

Keywords


[1] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421{1433.
[2] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-
Holland Math. Library 52, North-Holland, Amsterdam, 1994.
[3] J. Banks, Chaos for induced hyperspace maps, Chaos, Solitons and Fractals, 25 (2005), 681{
685.
[4] A.D. Barwell, C. Good, P. Oprocha and B.E. Raines, Characterizations of !-limit sets in
topologically hyperbolic systems, Discrete and Continuous Dynamical Systems, 33 (2013),
1819{1833.
[5] W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of
probability measures, Monatsh. Math., 79 (1975), 81{92.
[6] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew.
Math., 547 (2002), 51{68.
[7] C. Conley, Isolated invariant sets and the Morse index, Regional Conference Series in Math.,
No. 38, Amer. Math. Soc., Providence, RI, 1978.
[8] R.L. Devaney, An Introduction to Chaotic Dynamical Systems. Redwood City: Addison-
Wesley; 1989.
[9] L. Fernandez, C. Good, M. Puljiz and A. Ramrez, Chain transitivity in hyperspaces, Chaos,
Solitons and Fractals, 81 (2015), 83{90.
[10] L. Fernandez and C. Good, Shadowing for induced maps of hyperspaces, Fund. Math., 235
(2016), 277{286.
[11] J.L. Gomez-Rueda, A. Illanes and H. Mendez, Dynamics properties of the induced maps in
the symmetric products, Chaos, Solitons and Fractals, 45 (2012), 1180{1187.
[12] A. Illanes and S. B. Nadler Jr., Hyperspaces, Monographs and Textbooks in Pure and Applied
Mathematics, vol. 216, Marcel Dekker Inc., New York, 1999.
[13] A. Khan and P. Kumar, Recurrence and shadowing on induced map on hyperspaces, Far East
Journal of Dynamical Systems, 22 (2013), 1{16.
[14] S. Kolyada and L. Snoha, Some aspects of topological transitivity { A survey, Grazer Math.
Ber. (Bericht), 334 (1997), 3{35.
[15] J. Kupka, On Devaney chaotic induced fuzzy and set-valued dynamical systems, Fuzzy Sets
and Systems, 177 (2011), 34{44.
[16] J. Kupka, On fuzzifi cations of discrete dynamical systems, Information Sciences, 181 (2011),
2858{2872.
[17] J. Kupka, Some chaotic and mixing properties of fuzzifi ed dynamical systems, Information
Sciences, 279 (2014), 642{653.
[18] D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyper-
spaces, Chaos, Solitons and Fractals, 33 (2007), 76{86.
[19] J. Li, Equivalent conditions of Devaney chaos on the hyperspace, Journal of University of
Science and Technology of China, 44 (2014), 93{95.
[20] J. Li, P. Oprocha and X. Wu, Furstenberg families, sensitivity and the space of probability
measures, Nonlinearity, 30 (2017), 987{1005.
[21] J. Li, P. Oprocha, X. Ye and R. Zhang, When are all closed subsets recurrent?, Ergodic
Theory Dynam. Systems, 37 (2017), 2223{2254.
[22] J. Li, K. Yan and X. Ye, Recurrence properties and disjointness on the induced spaces,
Discrete and Continuous Dynamical Systems, 45 (2015), 1059{1073.

[23] G. Liao, L. Wang and Y. Zhang, Transitivity, mixing and chaos for a class of set valued
mappings, Sci. China Ser. A Math., 49 (2006), 1{8.
[24] P. Oprocha and X. Wu, On average tracing of periodic average pseudo orbits, Discrete and
Continuous Dynamical Systems, 37 (2017), 4943{4957.
[25] D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl.,
156 (2008), 251{261.
[26] H. Roman-Flores and Y. Chalco-Cano, Some chaotic properties of Zadeh's extension, Chaos,
Solitons and Fractals, 35 (2008), 452{459.
[27] H. Roman-Flores, Y. Chalco-Cano, G.N. Silva and J. Kupka, On turbulent, erratic and other
dynamical properties of Zadeh's extensions, Chaos, Solitons and Fractals, 44 (2011), 990{994.
[28] P. Sharman and A. Nagar, Inducing sensitivity on hyperspaces, Topology and its Applications,
157 (2010), 2052{2058.
[29] Y. Wang, G. Wei and W. Campbell, Sensitive dependence on initial conditions between
dynamical systems and their induced hyperspace dynamical systems, Topology and its Appli-
cations, 156 (2009), 803{811.
[30] Y. Wang and G. Wei, Dynamical systems over the space of upper semicontinuous fuzzy sets,
Fuzzy Sets and Systems, 209 (2012), 89{103.
[31] X. Wu, Chaos of transformations induced on the space of probability measures, Int. J. Bifur-
cation and Chaos, 26 (2016), 1650227 (12 pages).
[32] X. Wu, A remark on topological sequence entropy, Int. J. Bifurcation and Chaos, 27 (2017),
1750107 (7 pages).
[33] X. Wu and G. Chen, Sensitivity and transitivity of fuzzi fied dynamical systems, Information
Sciences, 396 (2017), 14{23.
[34] X. Wu, X. Ding, T. Lu and J. Wang, Topological dynamics of Zadeh's extension on upper
semi-continuous fuzzy sets, Int. J. Bifurcation and Chaos, 27 (2017), 1750165 (13 pages).
[35] X. Wu, P. Oprocha and G. Chen, On various defi nitions of shadowing with average error in
tracing, Nonlinearity, 29 (2016), 1942{1972.
[36] X. Wu and X. Wang, On the iteration properties of large deviations theorem, Int. J. Bifur-
cation and Chaos, 26 (2016), 1650054 (6 pages).
[37] X. Wu, J. Wang and G. Chen, F-sensitivity and multi-sensitivity of hyperspatial dynamical
systems, J. Math. Anal. Appl., 429 (2015), 16{26.
[38] X. Wu, X. Wang and G. Chen, On the large deviations theorem of weaker types, Int. J.
Bifurcation and Chaos, 27 (2017), 1750127 (12 pages).
[39] X. Wu, L. Wang and G. Chen, Weighted backward shift operators with invariant distribu-
tionally scrambled subsets, Ann. Funct. Anal., 8 (2017), 199{210.
[40] X. Wu, L. Wang and J. Liang, The chain properties and average shadowing property of
iterated function systems, Qual. Theory Dyn. Syst, 17 (2018), 219{227.
[41] Y. Wu and X. Xue, Shadowing property for induced set-valued dynamical systems of some
expansive maps, Dynam. Systems Appl., 19 (2010), 405{414.