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Keywords = Kato chaos

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9 pages, 271 KiB  
Article
Kato Chaos in Linear Dynamics
by Lixin Jiao, Lidong Wang and Heyong Wang
Mathematics 2023, 11(16), 3540; https://doi.org/10.3390/math11163540 - 16 Aug 2023
Cited by 1 | Viewed by 1355
Abstract
This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) [...] Read more.
This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics. Full article
15 pages, 308 KiB  
Article
Collective Sensitivity, Collective Accessibility, and Collective Kato’s Chaos in Duopoly Games
by Hongqing Wang, Tianxiu Lu, Risong Li, Yuanlin Chen, Yongjiang Li and Weizhen Quan
Mathematics 2022, 10(22), 4226; https://doi.org/10.3390/math10224226 - 12 Nov 2022
Cited by 1 | Viewed by 1230
Abstract
By using the uniform continuity of two onto maps, this paper further explores stronger forms of Kato’s chaos, sensitivity, and accessibility of Cournot maps. In particular, the sensitivity, the collective sensitivity, the accessibility, and the collective accessibility of the compositions of two reaction [...] Read more.
By using the uniform continuity of two onto maps, this paper further explores stronger forms of Kato’s chaos, sensitivity, and accessibility of Cournot maps. In particular, the sensitivity, the collective sensitivity, the accessibility, and the collective accessibility of the compositions of two reaction functions are studied. It is observed that a Cournot onto map H on a product space is sensitive (collectively sensitive, collectively accessible, accessible, or collectively Kato chaotic) if and only if the restriction of the map H2 to the MPE-set is sensitive as well. Several examples are given to show the necessity of the reaction functions being continuous onto maps. Full article
(This article belongs to the Special Issue Advances in Ergodic Theory and Its Applications)
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