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Some Iterative Properties of ( F 1 , F 2 ) -Chaos in Non-Autonomous Discrete Systems
Xiao Tang1, Guanrong Chen2, Tianxiu Lu3
1School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, China.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
This study extends chaos theory by introducing new definitions for scrambled sets in discrete systems. It establishes that these scrambled sets remain invariant under iterations, a key finding for dynamical systems research.
Area of Science:
- Dynamical Systems
- Chaos Theory
- Set Theory
Background:
- Li-Yorke chaos and distributional chaos are fundamental concepts in dynamical systems.
- Invariance of scrambled sets under iterations is crucial for understanding system behavior.
- Non-autonomous discrete systems present unique challenges for analyzing chaotic dynamics.
Purpose of the Study:
- To extend the concept of compound invariance of Li-Yorke chaos and distributional chaos.
- To introduce new definitions for (F1, F2)-scrambled sets in non-autonomous discrete systems.
- To investigate the properties P(k) and Q(k) of Furstenberg families.
Main Methods:
- Introduction of new definitions for (F1, F2)-scrambled sets tailored for non-autonomous systems.
- Formal definition and analysis of properties P(k) and Q(k) for Furstenberg families.
- Utilizing upper density of infinite subsets of positive integers (M-bar(s)) as a key parameter.
Main Results:
- Demonstration that Furstenberg family M-bar(s) possesses properties P(k) and Q(k) for any positive integer k and s in [0, 1].
- Established the equivalence of (M-bar(s), M-bar(t))-scrambled sets in (X, f1, infinity) and (X, f1, infinity[m]).
- Extended the understanding of invariance for scrambled sets under iterations in discrete dynamical systems.
Conclusions:
- The study provides a generalized framework for analyzing scrambled sets in non-autonomous discrete systems.
- The invariance property holds for these newly defined scrambled sets under specific iterative conditions.
- Findings contribute to a deeper comprehension of chaotic dynamics and set-theoretic properties in complex systems.
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