Shadowing properties and chaotic properties of non-autonomous product systems

Jingmin Pi; Tianxiu Lu; Jie Zhou; Jingmin Pi; Tianxiu Lu; Jie Zhou

doi:10.3934/math.20231021

AIMS Mathematics

2023, Volume 8, Issue 9: 20048-20062. doi: 10.3934/math.20231021

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Research article

Shadowing properties and chaotic properties of non-autonomous product 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: 15 April 2023 Revised: 31 May 2023 Accepted: 07 June 2023 Published: 16 June 2023
MSC : 37B45, 37B55, 54H20

This paper examines how properties such as shadowing properties, transitivity, and accessibility in non-autonomous discrete dynamical systems carry over to their product systems. The paper establishes a proof that the product system exhibits the pseudo-orbit shadowing property (PSP) if, and only if, both factor systems possess PSP. This relationship, which is both sufficient and necessary, also holds for the average shadowing property (ASP) and accessibility. Consequently, in practical problem scenarios, certain chaotic properties of two-dimensional systems can be simplified to those observed in one-dimensional systems. However, it should be noted that while the point-transitivity, transitivity, or mixing of the product system can be deduced from the factor systems, the reverse is not true. In particular, this paper constructs counterexamples to demonstrate that some of the theorems presented herein do not hold when considering their inverses.
- non-autonomous discrete dynamical systems,
- product map,
- shadowing properties,
- transitivity,
- accessibility
Citation: Jingmin Pi, Tianxiu Lu, Jie Zhou. Shadowing properties and chaotic properties of non-autonomous product systems[J]. AIMS Mathematics, 2023, 8(9): 20048-20062. doi: 10.3934/math.20231021

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Abstract

This paper examines how properties such as shadowing properties, transitivity, and accessibility in non-autonomous discrete dynamical systems carry over to their product systems. The paper establishes a proof that the product system exhibits the pseudo-orbit shadowing property (PSP) if, and only if, both factor systems possess PSP. This relationship, which is both sufficient and necessary, also holds for the average shadowing property (ASP) and accessibility. Consequently, in practical problem scenarios, certain chaotic properties of two-dimensional systems can be simplified to those observed in one-dimensional systems. However, it should be noted that while the point-transitivity, transitivity, or mixing of the product system can be deduced from the factor systems, the reverse is not true. In particular, this paper constructs counterexamples to demonstrate that some of the theorems presented herein do not hold when considering their inverses.

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