Distributionally n-chaotic dynamics for linear operators

Yin, Zongbin; Yang, Qigui

doi:10.1007/s13163-017-0226-5

Cited by 12 publications

(6 citation statements)

References 26 publications

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“…Concerning the last example, we would like to note that Z. Yin and Y. Huang have recently proved in [61] that for any open set U ⊆ (0, ∞) which is bounded away from zero, there exists a bounded linear operator T on l p , where 1 ≤ p < ∞, such that U = {λ > 0 : λT is distributionally chaotic}. Motivated by the results achieved in [61], for any integer i ∈ N 12 , any tuple r = (r 1 , r 2 , • • •, r N ) ∈ N N and any multivalued linear operator A on a Fréchet space X, we introduce the set…”

Section: Let Us Recall That For Any Mlo Extensionmentioning

confidence: 93%

“…Describing the structure of set DDC A,i, r is quite non-trivial and requires a series of further analyses. The existence of invariant (d, i)-distributionally scrambled sets and the existence of common (d, i)-distributionally irregular vectors for tuples of multivalued linear operators are delicate problems that will not be discussed here, as well (see [58]- [61] and references quoted therein for further information in this direction).…”

Section: Let Us Recall That For Any Mlo Extensionmentioning

confidence: 99%

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Disjoint distributional chaos in Fréchet spaces

Kostić¹

2018

Preprint

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We introduce several different notions of disjoint distributional chaos for sequences of multivalued linear operators in Fréchet spaces. Any of these notions seems to be new and not considered elsewhere even for linear continuous operators in Banach spaces. We focus special attention to the analysis of some specific classes of linear continuous operators having a certain disjoint distributionally chaotic behaviour, providing also a great number of illustrative examples and applications of our abstract theoretical results.

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Section: Let Us Recall That For Any Mlo Extensionmentioning

confidence: 93%

Section: Let Us Recall That For Any Mlo Extensionmentioning

confidence: 99%

Disjoint distributional chaos in Fréchet spaces

Kostić¹

2018

Preprint

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“…For operators on Banach spaces, DC1 and DC2 are always equivalent [13,Theorem 2], and imply Li-Yorke chaos. Li-Yorke chaos and distributional chaos for linear operators have been studied in [6,8,10,11,12,13,14,27,28,32,33,38,40,41,42], for instance.…”

Section: Notations and Preliminariesmentioning

confidence: 99%

Mean Li-Yorke chaos in Banach spaces

Bernardes

Bonilla

Peris

2020

Journal of Functional Analysis

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We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Cesàro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator T such that every nonzero vector is absolutely mean irregular for both T and T −1 . Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for C 0 -semigroups of operators on Banach spaces. 1 1 2010 Mathematics Subject Classification: 47A16, 37D45.

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“…This is partly because the rigorous proof of the existence of their chaotic behaviors is challenging. For more results on the chaos in infinite-dimensional dynamical systems, we refer to [7,15,21,22,23,24] and the references therein.…”

mentioning

confidence: 99%

“…Then by Theorem 2.10, the solutions w x (x, t) and w t (x, t) of system (23) have chaotic oscillations. Taking the parameter pairs (α 1 , β 1 ) = (0.1, 1) and (α 2 , β 2 ) = (0.8, 1), which satisfy (24), the spatiotemporal profiles of w x (x, t) and w t (x, t) of system (23) are shown in Figure 1. It is easy to see that w x (x, t) and w t (x, t) are rapidly oscillatory as t increases.…”

mentioning

confidence: 99%

Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

Yang¹,

Xiang²

2021

DCDS-S

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In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction.

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Distributionally n-chaotic dynamics for linear operators

Cited by 12 publications

References 26 publications

Disjoint distributional chaos in Fréchet spaces

Disjoint distributional chaos in Fréchet spaces

Mean Li-Yorke chaos in Banach spaces

Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

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