On chaotic and stable behaviour of the von Foerster–Lasota equation in some Orlicz spaces

Dawidowicz, Antoni Leon; Poskrobko, Anna

doi:10.3176/proc.2008.2.01

Cited by 10 publications

(7 citation statements)

References 13 publications

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“…We study chaoticity of the dynamical system in the sense of Devaney. In papers [5] and [6], we describe the asymptotic behaviour of the linear version of Lasota equation…”

Section: Dynamical Systemsmentioning

confidence: 99%

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On Chaos Behaviour of Nonlinear Lasota Equation in Lebesgue Space

Dawidowicz

Poskrobko

2020

J Dyn Control Syst

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We concern the asymptotic behaviour of the dynamical systems induced by nonlinear Lasota equation. We study chaoticity in the sense of Devaney and strong stability of the system. In many articles authors consider the properties of the linear version of the equation. By the construction of the operator in the separable space, we can formulate the relations between the solutions of the linear equation and its nonlinear version in Lebesgue space. The aim of the paper is to present the detailed construction of the operator thanks to which, the study of simple relationships allows determining the chaotic behaviour of the nonlinear equation.

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“…We study chaoticity of the dynamical system in the sense of Devaney. In papers [5] and [6], we describe the asymptotic behaviour of the linear version of Lasota equation…”

Section: Dynamical Systemsmentioning

confidence: 99%

“…In this section we consider the linear Lasota equation ( 3) with initial condition (2). In [5] and [6], it was proved that asymptotic behaviour of the dynamical system ( T t ) t≥0 in L p ([0, 1]), p > 0, space depends on the coefficient γ values. Its decisive value is − 1 p .…”

Section: Linear Lasota Equationmentioning

confidence: 99%

On Chaos Behaviour of Nonlinear Lasota Equation in Lebesgue Space

Dawidowicz

Poskrobko

2020

J Dyn Control Syst

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“…The dynamics associated to this equation has been also considered in other spaces: in Hölder spaces of continuous functions [59][60][61] and Orlicz spaces [62], and in Sobolev spaces of type…”

Section: Devaney Chaosmentioning

confidence: 99%

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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“…[2, Theorem 27]) while iv) follows from iii) by Theorem 2 and iv) trivially implies ii). For real valued h satisfying (13) it was shown by Dawidowicz and Poskrobko in [4] that T F,h is strongly stable whenever h(0) ≤ −1/p. Thus, there is a very strong dichotomie in the dynamical behavior of the von Foerster-Lasota semigroup on L p (0, 1).…”

Section: Continuing As Abovementioning

confidence: 99%

A remark on the Frequent Hypercyclicity Criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation

Kalmes

2015

Math. Nachr.

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We generalize a result for the translation C0‐semigroup on Lp(R+,μ) about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity Criterion due to Mangino and Peris to certain weighted composition C0‐semigroups. Such C0‐semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster–Lasota equation arising in mathematical biology. Weighted composition C0‐semigroups on Sobolev spaces are also considered.

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On chaotic and stable behaviour of the von Foerster–Lasota equation in some Orlicz spaces

Cited by 10 publications

References 13 publications

On Chaos Behaviour of Nonlinear Lasota Equation in Lebesgue Space

On Chaos Behaviour of Nonlinear Lasota Equation in Lebesgue Space

Linear dynamics of semigroups generated by differential operators

A remark on the Frequent Hypercyclicity Criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation

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