Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations

Barrachina, Xavier; Conejero, J. Alberto

doi:10.1155/2012/457019

Cited by 19 publications

(15 citation statements)

References 31 publications

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“…Furthermore, whenever the Desch-Schappacher-Webb criterion can be applied, we have that the C 0 -semigroup is not only Devaney chaotic but also distributionally chaotic, see [12,Cor. 31] and [7,Rem.3.8].…”

Section: Linear Dynamics Of C 0 -Semigroupsmentioning

confidence: 99%

Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation

et al. 2015

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The phenomenon of chaos has been exhibited in mathematical nonlinear models that describe traffic flows, see, for instance (Li and Gao in Modern At microscopic level, Devaney chaos and distributional chaos have been exhibited for some car-following models, such as the quick-thinking-driver model and the forward and backward control model (Barrachina et al. in 2015; Conejero et al. in Semigroup Forum, 2015). We present here the existence of chaos for the macroscopic model given by the Lighthill-Whitham-Richards equation.

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Section: Linear Dynamics Of C 0 -Semigroupsmentioning

confidence: 99%

Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation

et al. 2015

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“…for N = k(n k − m −k ), and we obtain that T is completely absolutely mean irregular as soon as we show that condition (2) in Theorem 35 is satisfied. For this, we notice that…”

Section: Generically Mean Li-yorke Chaotic Operatorsmentioning

confidence: 62%

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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“…Proceeding inductively we will get that x * i,n = 0 for i = 1, 2 and n ∈ N. We finally have x * = 0 and we conclude the result by applying Theorem 2.2. Finally, it is well known that distributional chaos holds whenever the DSW criterion can be applied [8,13]. Moreover, when DSW holds, the C 0 -semigroup admits a strongly mixing measure with full support on X ρ × X ρ , [35].…”

Section: )mentioning

confidence: 99%

Chaotic semigroups from second order partial differential equations

Conejero

Lizama

Murillo‐Arcila

2017

Journal of Mathematical Analysis and Applications

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Abstract. We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C 0 -semigroup corresponding to a class of second-order partial differential equations. We also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the C 0 -semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial differential equations, such as the telegraph equation, whose solutions satisfy these properties.

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Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations

Abstract: The notion of distributional chaos has been recently added to the study of the linear dynamics of operators andC0-semigroups of operators. We will study this notion of chaos for some examples ofC0-semigroups that are already known to be Devaney chaotic.

Cited by 19 publications

References 31 publications

Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation

Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation

Mean Li-Yorke chaos in Banach spaces

Chaotic semigroups from second order partial differential equations

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