Topological Chaos for a Class of Linear Models

Protopopescu, V.; Azmy, Yousry Y.

doi:10.1142/s0218202592000065

Cited by 44 publications

(34 citation statements)

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“…The first systematic study on hypercyclic semigroups was initiated by Desch, Schappacher and Webb in [20]. However, some examples and previous results were already obtained by that time, see for instance [21][22][23][24][25]. In that work, Desch et al introduced some computable conditions for hypercyclicity and Devaney chaos.…”

Section: Nd0mentioning

confidence: 99%

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Linear dynamics of semigroups generated by differential operators

et al. 2017

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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 .

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Section: Nd0mentioning

confidence: 99%

“…Protopopescu and Azmy introduced kinetic models that describe "death" process in the study of linear dynamics [23].…”

Section: Birth-and-death Modelsmentioning

confidence: 99%

Linear dynamics of semigroups generated by differential operators

et al. 2017

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“…The main purpose of this section is to present some results related to chaos for the evolution of birth-and-death type models with proliferation discussed in [7,11,44,58].…”

Section: State Of the Artmentioning

confidence: 99%

“…In [58], the authors studied the death part of the birth-and-death process. Following the same line, but allowing variable coefficients, Banasiak and Lachowicz [7] considered the following system of equations:…”

Section: Death Model With Variable Coefficientsmentioning

confidence: 99%

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Dynamics of strongly continuous semigroups associated to certain differential equations

Benlloch¹

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VALÈNCIA, Juliol de 2015 Als meus pares Ja saps que a tu iii AgradecimientosPodría empezar y acabar como decía Charles Bukowski en Post office (Cartero, 1971): "Esto se presenta como un libro de ficción y no está dedicado a nadie". En cambio, no podía dejar pasar la oportunidad de agradecer su tiempo y paciencia a toda esa gente que ha compartido conmigo este periodo de mi vida.En primer lugar, me gustaría agradecer de todo corazón el apoyo de mis directores de tesis Alfred Peris y Elisabetta Mangino, sin vosotros no hubiera sido posible. Por su atención, dedicación y por tener siempre la puerta abierta para mí. Ha sido un privilegio trabajar con ellos; me llevo muchas cosas en lo personal y profesional. También quería tener unas palabras para el profesor José Bonet, él sabe que esta aventura la empecé por él. No podía ignorar una llamada de una persona de su trayectoria para realizar uno de mis sueños, ser doctor en matemáticas, en parte, también se lo debía. Debo agradecerle esa oportunidad y los buenos ratos pasados en el IUMPA, fue una lástima que nuestros caminos se separaran.Como no, tengo que dar las gracias a todos los miembros del IUMPA y al Departament d'Anàlisi Matemàtica de la Universitat de València porque tuve un ambiente de trabajo envidiable, grandes profesionales y mejores personas. Máxime a Manolo, Domingo, Antonio Galbis, Carmina, Enrique Sánchez, Cala y David. Sus motivaciones y cercanía hacían todo más fácil. Mención especial para Minerva y Félix, por su paciencia con los doctorandos y sus gestiones administrativas (o malabares, según se mire). A Alberto por su cercanía, predisposición desinteresada y por lo que está por venir.No podían faltar mis compañeros de estudios. Que decir de Xavier Barrachina y Javi Falcó, los emigrantes del grupo, os deseo lo mejor y que volváis pronto y con trabajo. Orlando con el que disfruté mucho durante el máster, junto con Javi no he podido tener mejor compañía. A María José Beltrán, de las mejores personas que conozco, se lo merece todo y si se lo propone no tendrá techo. A Carme y Juanmi por las risas compartidas. A Marina, terremoto y brillantez a partes iguales, me has hecho pasar buenos ratos. A esas personas que han sido una parte importante de mi vida durante la carrera, ellas saben quién son. En especial a Quique, un hermano para mí, a Manuel y a Yassid sin los que mi camino por la universidad no hubiera sido igual.Quiero dar las gracias a todo el grupo de Terminal Polivalente de Castellón por acogerme en medio de mi tesis doctoral. En particular, a José María García y José Manuel Gimeno por darme la oportunidad de conocer el mundo naval. Gracias a su consideración tuve tiempo para acabar esta tesis compaginándola con el trabajo, en parte, también se lo debo. Por supuesto, a mis compañeros Nacho e Iñaki, creo que no podría tener mejor compañía. A Laura e Ignacio que me han echado un cable cuando se lo he pedido. En definitiva, a todos los compañeros. Esos "-¡Buenos días planner!... Buenos días resto de mortales." de David o esos "-¡Espabila!" de Jorg...

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Chaos for semigroups of unbounded operators

deLaubenfels¹,

Emamirad

Grosse‐Erdmann

2003

Mathematische Nachrichten

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In this paper we generalize the notion of hypercyclic and chaotic semigroups to families of unbounded operators. We study this concept within the frameworks of C-regularized semigroups and of regular distribution semigroups. We then apply our results to unbounded semigroups generated by differential operators with constant coefficients in weighted spaces and to the unbounded semigroup {(−∆) t } t≥0 , where ∆ is the Laplacian operator.

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Topological Chaos for a Class of Linear Models

Cited by 44 publications

References 0 publications

Linear dynamics of semigroups generated by differential operators

Linear dynamics of semigroups generated by differential operators

Dynamics of strongly continuous semigroups associated to certain differential equations

Chaos for semigroups of unbounded operators

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