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On fuzzifications of discrete dynamical systems. (English) Zbl 1229.93107

Summary: Let \(X\) denote a locally compact metric space and \(\varphi:X \rightarrow X\) be a continuous map. In the 1970s, Zadeh presented an extension principle helping us to fuzzify the dynamical system \((X,\varphi)\), i.e., to obtain a map \(\Phi\) for the space of fuzzy sets on \(X\). We extend an idea mentioned in [P. Diamond and A. Pokrovskii, Fuzzy Sets Syst. 61, No. 3, 277–283 (1994; Zbl 0827.58037)] to generalize Zadeh’s original extension principle.
In this paper, we study basic properties of so-called \(g\)-fuzzifications, such as their continuity properties. We also show that, for any \(g\)-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on \(X\) induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems.
Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, the endograph and levelwise topologies.

MSC:

93C42 Fuzzy control/observation systems
93C25 Control/observation systems in abstract spaces

Citations:

Zbl 0827.58037
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References:

[1] Allahviranloo, T.; Shafiee, M.; Nejatbakhsh, Y., A note on fuzzy differential equations and the extension principle, Inform. sci., 179, 2049-2051, (2009) · Zbl 1177.34014
[2] Agiza, H.N.; Elsadany, A.A., Chaotic dynamics in nonlinear duopoly game next term with heterogeneous players, Appl. math. comput., 149, 843-860, (2004) · Zbl 1064.91027
[3] Akin, E., The general topology of dynamical systems, Graduate studies in mathematics, vol. 1, (1993), American Mathematical Society Providence, RI · Zbl 0781.54025
[4] L.C. de Barros, R.C. Bassanezi, P.A. Tonelli, On the continuity of The Zadeh’s extension, in: Proc. IFSA’97 Congress, Prague.
[5] Bassanezi, R.C.; de Barros, L.C.; Tonelli, P.A., Attractors and asymptotic stability for fuzzy dynamical systems, Fuzzy sets syst., 113, 473-483, (2000) · Zbl 0954.37022
[6] Bělohlávek, R., A note on the extension principle, J. math. anal. appl., 248, 678-682, (2000) · Zbl 0965.03067
[7] Block, L.S.; Coppel, W.A., One-dimensional dynamics, Lecture notes in mathematics, vol. 1513, (1992), Springer-Verlag Berlin
[8] Brannstrom, A.; Sumpter, D.J., The role of competition and clustering in population dynamics, Proc. biol. sci., 272, 2065-2072, (2005)
[9] Canovas, J.S.; Linero, A., Topological dynamic classification of duopoly games, Chaos solitons fract., 12, 1259-1266, (2001) · Zbl 1035.91018
[10] Chalco-Cano, Y.; Jiménez-Gamero, M.; Román-Flores, H.; Rojas-Medar, M.A., An approximation to the extension principle using decomposition of fuzzy intervals, Fuzzy sets syst., 159, 3245-3258, (2008) · Zbl 1176.03025
[11] Chalco-Cano, Y.; Roman-Flores, H.; Rojas-Medar, M.; Saavedra, O.R.; Jimenez-Gamero, M.D., The extension principle and a decomposition of fuzzy sets, Inform. sci., 177, 5394-5403, (2007) · Zbl 1124.03326
[12] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, (1994), World Scientific · Zbl 0843.54041
[13] Diamond, P.; Kloeden, P.; Pokrovskii, A., Absolute retracts and a general fixed point theorem for fuzzy sets, Fuzzy sets syst., 86, 377-380, (1997) · Zbl 0917.54045
[14] Diamond, P.; Pokrovskii, A., Chaos, entropy and a generalized extension principle, Fuzzy sets syst., 61, 277-283, (1994) · Zbl 0827.58037
[15] Qiu, Dong; Shu, Lan, Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings, Inform. sci., 178, 3595-3604, (2008) · Zbl 1151.54010
[16] Dubois, D.; Prade, H., Operation on fuzzy number, Int. J. system sci., 9, 613-626, (1978) · Zbl 0383.94045
[17] Franaszek, M.; Simiu, E., Noise-induced snap-through of a buckled column with continuously distributed mass: a chaotic dynamics approach, Int. J. non-linear mech., 31, 861-869, (1996) · Zbl 0889.73050
[18] Gerla, G.; Scarpati, L., Extension principles for fuzzy set theory, Inform. sci., 106, 49-69, (1998) · Zbl 0929.03055
[19] Ghil, B.M., On the convergence of fuzzy-number valued functions, Fuzzy sets syst., 87, 373-375, (1997) · Zbl 0924.26015
[20] Garcı´a Guirao, J.L.; Kwietniak, D.; Lampart, M.; Oprocha, P.; Peris, A., Chaos on hyperspaces, Nonlinear anal. theory, methods appl., 71, 1-8, (2009) · Zbl 1175.37024
[21] Hanss, M., Applied fuzzy arithmetic: an introduction with engineering applications, (2005), Springer-Verlag Berlin · Zbl 1085.03041
[22] Hiew, Hong Liang; Tsang, Chi Ping, An adaptive fuzzy system for modeling chaos, Inform. sci., 81, 193-212, (1994) · Zbl 0827.58039
[23] Kloeden, P.E., Fuzzy dynamical systems, Fuzzy sets and systems, 7, 275-296, (1982) · Zbl 0509.54040
[24] Kuratowski, K., Topology, Vol. II, (1968), Academic Press London, New York
[25] J. Kupka, Some chaotic and mixing properties of Zadeh’s extension, in: Proceedings of IFSA World Congress/EUSFLAT Conference, Universidade Tecnica de Lisboa, Lisabon, Portugalsko, 2009, pp. 589-594.
[26] Kyrtsou, C.; Labys, W., Evidence for chaotic dependence between US inflation and commodity prices, J. macroecon., 28, 256-266, (2006)
[27] Nguyen, H.F., A note on the extension principles for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004
[28] Vazquez-Medina, R.; Diaz-Mendez, A.; del Rio-Correa, J.L.; Lopez-Hernandez, J., Design of chaotic term analog noise generators with logistic map and MOS QT circuits, Chaos solitons fract., 40, 1779-1793, (2009) · Zbl 1198.94209
[29] Pederson, S.M., Fuzzy homoclinic orbits and commuting fuzzifications, Fuzzy sets syst., 155, 361-371, (2005), vol. 3 · Zbl 1093.37007
[30] Peris, A., Set – valued discrete chaos, Chaos solitons fract., 26, 19-23, (2005) · Zbl 1079.37024
[31] Román-Flores, H.; Barros, L.C.; Bassanezi, R.C., A note on zadeh’s extensions, Fuzzy sets syst., 117, 327-331, (2001) · Zbl 0968.54007
[32] Román-Flores, H.; Chalco-Cano, Y., Some chaotic properties of zadeh’s extension, Chaos solitons fract., 35, 452-459, (2008) · Zbl 1142.37308
[33] Sevastjanov, P.; Dymova, L., A new method for solving interval and fuzzy equations: linear case, Inform. sci., 179, 925-937, (2009) · Zbl 1160.65010
[34] Sharkovsky, A.N.; Kolyada, S.F.; Sivak, A.G.; Fedorenko, V.V., Dynamics of one-dimensional maps, (1997), Kluwer Academic Publishers · Zbl 0881.58020
[35] Yager, R.R., Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures, Inform. sci., 178, 3565-3576, (2008) · Zbl 1154.68530
[36] Zadeh, L., Toward a generalized theory of uncertainty (GTU) an outline, Inform. sci., 172, 1-40, (2005) · Zbl 1074.94021
[37] Zadeh, L., Is there a need for fuzzy logic?, Inform. sci., 178, 2751-2779, (2008) · Zbl 1148.68047
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