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Some chaotic properties of fuzzified dynamical systems
Cuina Ma1, Peiyong Zhu1, Tianxiu Lu2
1School of Mathematical Sciences, University of Electronic Science and Technology of China, 611731 Chengdu, People's Republic of China.
Springerplus
|June 23, 2016
Summary
The study explores the behavior of continuous maps on fuzzy sets. It demonstrates that the g-fuzzification of a system exhibits turbulence or erraticsm if the original system does, and shows expansiveness is preserved under g-fuzzification.
Area of Science:
- Dynamical Systems
- Fuzzy Set Theory
- Topology
Background:
- Continuous maps on compact metric spaces are fundamental in dynamical systems.
- Fuzzy set theory provides tools to model uncertainty and vagueness.
- The levelwise topology on the space of fuzzy sets is a key structure for analysis.
Purpose of the Study:
- To investigate the relationship between the dynamical properties of a continuous map and its g-fuzzification.
- To determine how concepts like turbulence, erraticsm, and expansiveness translate to the fuzzy set space.
Main Methods:
- Considered a compact metric space X and a continuous map f: X -> X.
- Utilized the space of all nonempty fuzzy sets on X with the levelwise topology.
- Analyzed the g-fuzzification of the system f.
Main Results:
- Established that the g-fuzzification of f is turbulent or erratic if and only if f itself is turbulent or erratic.
- Proved that f is [Formula: see text]-expansive if and only if its g-fuzzification is [Formula: see text]-expansive.
Conclusions:
- The g-fuzzification process preserves the turbulent, erratic, and [Formula: see text]-expansive properties of the original dynamical system.
- This work extends the understanding of dynamical systems to the realm of fuzzy sets.
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