Chaos, entropy and a generalized extension principle. (English) Zbl 0827.58037

Diamond, Phil; Pokrovskii, Aleksej

Fuzzy Sets Syst. 61, No. 3, 277-283 (1994).

Summary: We show a relationship between a very simple criterion, positive topological entropy and Li-Yorke chaos. A general definition of fuzzification and level set, based on \(t\)-norms/conorms and their diagonal functions, is introduced. The chaos theorem of Benhabib and Day for set valued mappings is considerably strengthened and generalised.

MSC:

37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
94D05	Fuzzy sets and logic (in connection with information, communication, or circuits theory)
54C70	Entropy in general topology
03E72	Theory of fuzzy sets, etc.

Keywords:

dynamical system; topological entropy; Li-Yorke chaos; fuzzification

References:

[1]	Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On Devaney’s definition of chaos, Amer. Math. Monthly, 99, 332-334 (1992) · Zbl 0758.58019
[2]	Benhabib, J.; Day, R. H., Rational choice and erratic behaviour, Rev. Econ. Stud., 48, 459-471 (1981) · Zbl 0463.90006
[3]	Blank, M. L., Small perturbations of chaotic dynamical systems, Russian Math. Surveys, 44, 6, 1-33 (1989) · Zbl 0702.58063
[4]	Bowen, R., Topological entropy and Axiom A, (Proc. of Symposia in Pure Mathematics, Volume XIV (1970), American Mathematical Society: American Mathematical Society Providence, Rhode Island) · Zbl 0207.54402
[5]	Debreu, G., Integration of correspondences, (Proc. Fifth Berkeley Symp. Math. Statist. Probability, Vol. II (1967)), 351-372 · Zbl 0211.52803
[6]	Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1989), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0695.58002
[7]	Diamond, P., Chaotic behaviour of systems of difference equations, Intern. J. System Sci., 7, 953-956 (1976) · Zbl 0336.93004
[8]	Diamond, P., Chaos and fuzzy representations of dynamical systems, (Proc. 2nd Int. Conf. on Fuzzy Logic & Neural Networks. Proc. 2nd Int. Conf. on Fuzzy Logic & Neural Networks, Iizuka, Japan, July 17-22, 1992 (1992), FLSI: FLSI Iizuka, Japan), 51-58
[10]	Eckmann, J.-P.; Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617-656 (1985) · Zbl 0989.37516
[11]	Fullér, R.; Kereszfalvi, T., On generalization of Nguyen’s theorem, Fuzzy Sets and Systems, 41, 371-374 (1991) · Zbl 0755.04004
[12]	Gupta, M. M.; Qi, J., Design of fuzzy logic controllers based on generalized T-operators, Fuzzy Sets and Systems, 40, 473-489 (1991) · Zbl 0732.93050
[13]	Kloeden, P., Compact supported endographs and fuzzy sets, Fuzzy Sets and Systems, 4, 193-201 (1980) · Zbl 0441.54008
[14]	Kloeden, P., Chaotic difference equations in \(R^n\), J. Australian Math. Soc. A, 31, 217-225 (1981) · Zbl 0471.39001
[15]	Li, T. Y.; Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82, 985-992 (1975) · Zbl 0351.92021
[16]	Marotto, F. R., Snap-back repellers imply chaos in \(R^n\), J. Math. Anal. Appl., 63, 199-223 (1978) · Zbl 0381.58004
[17]	Nguyen, H. T., A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64, 369-380 (1978) · Zbl 0377.04004
[18]	Teodorescu, H. N., Chaos in fuzzy systems and signals, (Proc. 2nd Int. Conf. on Fuzzy Logic & Neural Networks. Proc. 2nd Int. Conf. on Fuzzy Logic & Neural Networks, Iizuka, Japan, July 17-22, 1992 (1992), FLSI: FLSI Iizuka, Japan), 21-50
[19]	Wiggins, S., Global Bifurcations and Chaos (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0661.58001