Distributional Chaos in Random Dynamical Systems
Jozef Kov\'a\v{c}, Katar\'ina Jankov\'a
TL;DR
This paper introduces the concept of distributional chaos in random dynamical systems, providing conditions for chaos, examples, and analyzing the stability of chaos under system variations.
Contribution
It defines distributional chaos for random systems, explores its properties, and shows that chaos can arise from nonchaotic functions, highlighting its instability.
Findings
Chaotic systems can be generated by nonchaotic functions.
Distributional chaos can be unstable under system perturbations.
Provides sufficient conditions for zero measure of chaos.
Abstract
In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e., a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.
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