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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
