Distributional chaos for strongly continuous semigroups of operators

Albanese, Angela A.; Barrachina, Xavier; Mangino, Elisabetta M.; Peris, Alfredo

doi:10.3934/cpaa.2013.12.2069

Cited by 30 publications

(45 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was incorporated to the setting of linear dynamics in [81,82], and thoroughly studied in [83]. The corresponding version for C 0 -semigroups was given in [84]. In this notion we require a scrambled set S such that the orbits of any couple of distinct points are closed enough or at least at some distance, measured in terms of upper densities.…”

Section: Distributional Chaosmentioning

confidence: 99%

“…It was shown in [83] that the existence of distributionally irregular vectors was equivalent to distributional chaos for single operators, and later generalized for C 0 -semigroups in [84].…”

Section: Definition 110 ([84])mentioning

confidence: 99%

“…Concerning distributional chaos, some criteria for C 0 -semigroups were provided in [84], inspired by the ones given for single operators in [83]. We recall the following useful one: …”

Section: The Role Of the Infinitesimal Generatormentioning

confidence: 99%

See 2 more Smart Citations

Linear dynamics of semigroups generated by differential operators

et al. 2017

Self Cite

View full text Add to dashboard Cite

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C 0 -semigroups of linear and continuous operators.We will review some of these notions and we will discuss basic properties of the dynamics of C 0 -semigroups. We will also study in detail the dynamics of the translation C 0 -semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C 0 -semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models. . On the one hand, it is connected with the still unsolved Invariant Subspace Problem on Hilbert spaces, which asks for the existence of an operator with no nontrivial closed invariant subset. The answer was positive in Banach spaces, as it was seen by Read [4] in`1. On the other hand, there is a number of different connections of linear dynamics with different areas such as algebra, topology, real and complex analysis, functional analysis, approximation theory, number theory, and probability.The advances in the area were first compiled by Grosse-Erdmann [5,6]. The monographs of Bayart and Matheron [7] and Grosse-Erdmann and Peris [8] represent a good source of the state of the art in the area. The study of the size and the algebraic structure of the set of vectors with wild behaviour, provides an interesting field for continuing with the quest of surprising examples of sets of pathological elements that, however, are preserved by the elementary algebraic operators. In this line, a selection of topics was recently revisited by Aron et al. in [9, Ch. 4], see also [10].Given a family fT i g i 2I of linear and continuous operators on an infinite-dimensional separable Banach space X , we say that it is universal if there exists some element x 2 X such that, its orbit fT i x W i 2 I g is dense in X .

show abstract

Section: Distributional Chaosmentioning

confidence: 99%

Section: Definition 110 ([84])mentioning

confidence: 99%

See 1 more Smart Citation

Linear dynamics of semigroups generated by differential operators

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…We will study the existence of distributional chaos for the solutions to this general model, and we will analyze, as a particular case, its consequences for the (IFBC) car-following model. In order to do this, we will use the following criterion which ensures distributional chaos and can be found in [4]. …”

Section: Chaos For the Forward And Backward Modelmentioning

confidence: 99%

“…Nevertheless, for the case when a, b, d where real numbers, the same result holds just taking the restriction of W 0 to the real numbers. For further details we refer the reader to the proof of Criterion 3.2, see [4,12,36]. This approach can be compared with [24,Th.…”

Section: Lemma 33 (See Lemma 1 In [10])mentioning

confidence: 99%

Distributional chaos for the Forward and Backward Control traffic model

Barrachina

Conejero

Murillo‐Arcila

et al. 2015

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

Abstract. The interest in car-following models has increased in the last years due to its connection with vehicle-to-vehicle communications and the development of driverless cars. Some non-linear models such as the Gazes-Herman Rothery model were already known to be chaotic. We consider the linear Forward and Backward Control traffic model for an infinte number of cars on a track. We show the existence of solutions with a chaotic behaviour by using some results of linear dynamics of C0-semigroups. In contrast, we also analyze which initial configurations lead to stable solutions.

show abstract

The Specification Property for $$C_0$$-Semigroups

et al. 2019

View full text Add to dashboard Cite

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, C 0 -semigroups. In addition, we study the relationships of the specification property for C 0 -semigroups (SgSP) with other dynamical properties: mixing, Devaney's chaos, distributional chaos and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black-Scholes equation.

show abstract

Distributional chaos for strongly continuous semigroups of operators

Cited by 30 publications

References 30 publications

Linear dynamics of semigroups generated by differential operators

Linear dynamics of semigroups generated by differential operators

Distributional chaos for the Forward and Backward Control traffic model

The Specification Property for $$C_0$$-Semigroups

Contact Info

Product

Resources

About