Chaotic iterations of fuzzy sets. (English) Zbl 0746.54010
The author gives sufficient conditions for a continuous mapping from a space of fuzzy sets into itself to be chaotic. The conditions are illustrated by an example involving fuzzy sets defined on \(\mathbb{R}\).
Reviewer: O.Kaleva (Tampere)
MSC:
54E40 | Special maps on metric spaces |
54B20 | Hyperspaces in general topology |
37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
03E72 | Theory of fuzzy sets, etc. |
37B99 | Topological dynamics |
54E50 | Complete metric spaces |
References:
[1] | Diamond, P., Fuzzy chaos (1977), Mathematics Department, University of Queensland, Preprint |
[2] | Diamond, P.; Kloeden, P. E., Characterization of compact subsets of fuzzy sets, Fuzzy Sets and Systems, 35, 241-249 (1990) · Zbl 0704.54006 |
[3] | Kaleva, O., On the convergence of fuzzy sets, Fuzzy Sets and Systems, 17, 53-65 (1985) · Zbl 0584.54004 |
[4] | Kloeden, P. E., Chaotic difference equations in \(R^n\), J. Austral. Math. Soc. Ser. A, 31, 217-225 (1981) · Zbl 0471.39001 |
[5] | Kloeden, P. E., Cycles and chaos in higher dimensional difference equations, (Mitropolsky, Yu. A., Proc. 9th Internat. Conf. Nonlinear Oscillations, Vol. 2 (1984), Kiev Naukova Dumka), 184-187 |
[6] | Kloeden, P. E.; Mees, A. I., Chaotic phenomena, Bull. Math. Biol., 47, 697-738 (1985) · Zbl 0586.92002 |
[7] | Li, T. Y.; Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82, 985-992 (1975) · Zbl 0351.92021 |
[8] | May, R. M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467 (1976) · Zbl 1369.37088 |
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