Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems

Li, Risong; Li, Yongjiang; Lu, Tianxiu; Zhao, Jiazheng; Su, Jing

doi:10.3390/e26080674

Open AccessArticle

Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems

by

Risong Li

^1,†

,

Yongjiang Li

^1,†,

Tianxiu Lu

^2,*

,

Jiazheng Zhao

^2,3 and

Jing Su

¹

School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang 524025, China

²

College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China

³

South Sichuan Applied Mathematics Research Center, Zigong 643000, China

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Entropy 2024, 26(8), 674; https://doi.org/10.3390/e26080674

Submission received: 14 May 2024 / Revised: 8 July 2024 / Accepted: 6 August 2024 / Published: 9 August 2024

(This article belongs to the Section Complexity)

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Abstract

Assume that

(Y, ρ)

is a nontrivial complete metric space, and that

(Y, g_{1, \infty})

is a time-varying discrete dynamical system (T-VDDS), which is given by sequences

{(g_{l})}_{l = 1}^{\infty}

of continuous selfmaps

g_{l} : Y \to Y

. In this paper, for a given Furstenberg family

G

and a given T-VDDS

(Y, g_{1, \infty})

,

G

-scrambled pairs of points of the system

(Y, g_{1, \infty})

(which contains the well-known scrambled pairs) are provided. Some properties of the set of

G

-scrambled pairs of a given T-VDDS

(Y, g_{1, \infty})

are studied. Moreover, the generically

G

-chaotic T-VDDS and the generically strongly

G

-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly

G

-chaotic is also presented.

Keywords: Furstenberg family; scrambled pair; scrambled set; generically

G

-scrambled time-varying discrete dynamical system

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MDPI and ACS Style

Li, R.; Li, Y.; Lu, T.; Zhao, J.; Su, J. Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems. Entropy 2024, 26, 674. https://doi.org/10.3390/e26080674

AMA Style

Li R, Li Y, Lu T, Zhao J, Su J. Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems. Entropy. 2024; 26(8):674. https://doi.org/10.3390/e26080674

Chicago/Turabian Style

Li, Risong, Yongjiang Li, Tianxiu Lu, Jiazheng Zhao, and Jing Su. 2024. "Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems" Entropy 26, no. 8: 674. https://doi.org/10.3390/e26080674

APA Style

Li, R., Li, Y., Lu, T., Zhao, J., & Su, J. (2024). Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems. Entropy, 26(8), 674. https://doi.org/10.3390/e26080674

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems

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