The expansivity and sensitivity of the set-valued discrete dynamical systems

Jie Zhou; Tianxiu Lu; Jiazheng Zhao; Jie Zhou; Tianxiu Lu; Jiazheng Zhao

doi:10.3934/math.20241171

AIMS Mathematics

2024, Volume 9, Issue 9: 24089-24108. doi: 10.3934/math.20241171

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Research article Special Issues

The expansivity and sensitivity of the set-valued discrete dynamical systems

1.
College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China
2.
South Sichuan Applied Mathematics Research Center, Zigong 643000, China

Received: 18 June 2024 Revised: 29 July 2024 Accepted: 05 August 2024 Published: 14 August 2024
MSC : 37D45, 37B40, 54H20

Let $(X, d)$ be a metric space and $\mathcal{H}(X)$ represent all non-empty, compact subsets of $X$. The expansivity of the multivalued map sequence $\bar{f}_{1, \infty}: \mathcal{H}(X) \to \mathcal{H}(X)$, including expansivity, positive $\aleph_0$-expansivity, were investigated. Also, stronger forms of sensitivities, such as multi-sensitivity and syndetical sensitivity, were explored. This research demonstrated that some chaotic properties can be mutually derived between $(f_{1, \infty}, X)$ and $(\bar{f}_{1, \infty}, \mathcal{H}(X))$, showing fundamental similarities between these systems. Conversely, the inability to derive other properties underlined essential differences between them. These insights are crucial for simplifying theoretical models and enhancing independent research. Lastly, the relationship between expansivity and sensitivity was discussed and the concept of topological conjugacy to the system $ (\bar{f}_{1, \infty}, \mathcal{H}(X)) $ was extended.
- non-autonomous discrete dynamical systems,
- expansivity,
- sensitivity,
- topological conjugacy
Citation: Jie Zhou, Tianxiu Lu, Jiazheng Zhao. The expansivity and sensitivity of the set-valued discrete dynamical systems[J]. AIMS Mathematics, 2024, 9(9): 24089-24108. doi: 10.3934/math.20241171

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Abstract

Let $(X, d)$ be a metric space and $\mathcal{H}(X)$ represent all non-empty, compact subsets of $X$. The expansivity of the multivalued map sequence $\bar{f}_{1, \infty}: \mathcal{H}(X) \to \mathcal{H}(X)$, including expansivity, positive $\aleph_0$-expansivity, were investigated. Also, stronger forms of sensitivities, such as multi-sensitivity and syndetical sensitivity, were explored. This research demonstrated that some chaotic properties can be mutually derived between $(f_{1, \infty}, X)$ and $(\bar{f}_{1, \infty}, \mathcal{H}(X))$, showing fundamental similarities between these systems. Conversely, the inability to derive other properties underlined essential differences between them. These insights are crucial for simplifying theoretical models and enhancing independent research. Lastly, the relationship between expansivity and sensitivity was discussed and the concept of topological conjugacy to the system $ (\bar{f}_{1, \infty}, \mathcal{H}(X)) $ was extended.

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