zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Chaos, entropy and a generalized extension principle. (English) Zbl 0827.58037
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.

37D45Strange attractors, chaotic dynamics
94D05Fuzzy sets and logic in connection with communication
54C70Topological entropy
03E72Fuzzy set theory
Full Text: DOI
[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, No. 6, 1-33 (1989) · Zbl 0702.58063
[4] Bowen, R.: Topological entropy and axiom A. Proc. of symposia in pure mathematics (1970) · Zbl 0207.54402
[5] Debreu, G.: Integration of correspondences. Proc. fifth Berkeley symp. Math. statist. Probability, 351-372 (1967) · Zbl 0211.52803
[6] Devaney, R. L.: An introduction to chaotic dynamical systems. (1989) · 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, 51-58 (1992)
[9] P. Diamond, P. Kloeden and A. Pokrovskii, Absolute retracts and a general fized point theorem for fuzzy sets, to appear in Fuzzy Sets and Systems. · Zbl 0917.54045
[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 rn. 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 rn. 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, 21-50 (1992)
[19] Wiggins, S.: Global bifurcations and chaos. (1988) · Zbl 0661.58001