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On Devaney chaotic induced fuzzy and set-valued dynamical systems. (English) Zbl 1242.37014
Summary: It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e., a discrete dynamical system on the space of fuzzy sets. In this paper we study relations between dynamical properties of the original and the fuzzified dynamical system. Especially, we study conditions used in the definition of Devaney chaotic maps, i.e., periodic density and transitivity. Among other things we show that dynamical behavior of the set-valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification. This paper contains the solution of the problem that was partially solved by H. Román-Flores and Y. Chalco-Cano [Chaos Solitons Fractals 35, No. 3, 452–459 (2008; Zbl 1142.37308)].
MSC:
37B99Topological dynamics
37D45Strange attractors, chaotic dynamics
37C25Fixed points, periodic points, fixed-point index theory
54A40Fuzzy topology
References:
[1]Acosta, G.; Illanes, A.; Méndez-Lango, H.: The transitivity of induced maps, Topology and its applications 159, 1013-1033 (2009)
[2]Balibrea, F.; Snoha, L.: Topological entropy of devaney chaotic maps, Topology and its applications 133, 225-239 (2003) · Zbl 1026.37002 · doi:10.1016/S0166-8641(03)00090-7
[3]Banks, J.: Chaos for induced hyperspace maps, Chaos, solitons and fractals 25, 681-685 (2005) · Zbl 1071.37012 · doi:10.1016/j.chaos.2004.11.089
[4]Bauer, W.; Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures, Monatshefte mathematics 79, 81-92 (1975) · Zbl 0314.54042 · doi:10.1007/BF01585664
[5]Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P.: On devaney’s definition of chaos, American mathematics monthly 99, 332-334 (1992) · Zbl 0758.58019 · doi:10.2307/2324899
[6]Cánovas, J. S.; Kupka, J.: Topological entropy of fuzzified dynamical systems, Fuzzy sets and systems 165, No. 1, 37-49 (2011)
[7]Devaney, R.: An introduction to chaotic dynamical systems, (2003) · Zbl 1025.37001
[8]Diamond, P.; Pokrovskii, A.: Chaos, entropy and a generalized extension principle, Fuzzy sets and systems 61, 277-283 (1994) · Zbl 0827.58037 · doi:10.1016/0165-0114(94)90170-8
[9]Gengrong, Z.; Fanping, Z.; Xinhe, L.: Devaney’s chaotic on induced maps of hyperspace, Chaos, solitons and fractals 27, 471-475 (2006)
[10]Glasner, E.; Weiss, B.: Sensitive dependence on initial conditions, Nonlinearity 6, 1067-1075 (1993) · Zbl 0790.58025 · doi:10.1088/0951-7715/6/6/014
[11]Guirao, J. L. García; Kwietniak, D.; Lampart, M.; Oprocha, P.; Peris, A.: Chaos on hyperspaces, Nonlinear analysis: theory, methods and applications 71, No. 1 – 2, 1-8 (2009)
[12]Forti, G. L.: Various notions of chaos for discrete dynamical systems. A brief survey, Aequationes mathematics 70, No. 1 – 2, 1-13 (2005) · Zbl 1080.37010 · doi:10.1007/s00010-005-2771-0
[13]Klein, E.; Thompson, A.: Theory of correspondences, (1984)
[14]Kolyada, S. F.: Li-Yorke sensitivity and other concepts of chaos, Ukrainskii matematicheskii zhurnal 56, No. 8, 1043-1061 (2004) · Zbl 1075.37500 · doi:10.1007/s11253-005-0055-4
[15]Kupka, J.: On fuzzifications of discrete dynamical systems, Information sciences 181, 2858-2872 (2011) · Zbl 1229.93107 · doi:10.1016/j.ins.2011.02.024
[16]J. Kupka, Some chaotic and mixing properties of fuzzified dynamical systems, Information Sciences, submitted for publication.
[17]Kuratowski, K.: Topology, Topology (1968)
[18]Kwietniak, D.; Misiurewicz, M.: Exact devaney chaos and entropy, Qualitative theory of dynamical systems 6, No. 1, 169-179 (2005) · Zbl 1119.37027 · doi:10.1007/BF02972670
[19]Pederson, S. M.: Fuzzy homoclinic orbits and commuting fuzzifications, Fuzzy sets and systems 155, No. 3, 361-371 (2005) · Zbl 1093.37007 · doi:10.1016/j.fss.2005.05.007
[20]Peris, A.: Set-valued discrete chaos, Chaos, solitons and fractals 26, 19-23 (2005) · Zbl 1079.37024 · doi:10.1016/j.chaos.2004.12.039
[21]Román-Flores, H.: A note on transitivity in set-valued discrete systems, Chaos, solitons and fractals 17, 99-104 (2003) · Zbl 1098.37008 · doi:10.1016/S0960-0779(02)00406-X
[22]Román-Flores, H.; Chalco-Cano, Y.: Some chaotic properties of zadeh’s extension, Chaos, solitons and fractals 35, No. 3, 452-459 (2008) · Zbl 1142.37308 · doi:10.1016/j.chaos.2006.05.036
[23]Román-Flores, H.; Chalco-Cano, Y.: Robinson’s chaos in set-valued discrete systems, Chaos, solitons and fractals 25, 33-42 (2005) · Zbl 1071.37013 · doi:10.1016/j.chaos.2004.11.006
[24]Xiong, J. C.; Yang, Z. G.: Chaos caused by a topologically mixing map. Dynamical systems and related topics, (1991)
[25]Zadeh, L.: Fuzzy sets, Information and control 8, 338-353 (1965) · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X