Wyniki wyszukiwania - Biblioteka Nauki

1

Algorithmic Approach to Devaney Chaos in Shift Spaces

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Oprocha P.

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2008

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tom Vol. 87, nr 3-4

435-446

EN

Among the class of sofic shifts Devaney chaos is equivalent to topological transitivity. The aim of this paper is to study possibilities of detection of this phenomena when a right-resolving graph presentation of a sofic shift is given. We show that Devaney chaos detection is co-NP-hard problem and point out some possible improvements of algorithms known from the literature.

2

Mieszanie topologiczne w dyskretnych układach dynamicznych - krótkie wprowadzenie

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Oprocha P.

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2008

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tom T. 12, z. 1

77-88

PL

Badając dynamikę procesów zmieniających się w czasie, zazwyczaj mamy do czynienia z dwoma głównymi kierunkami badań: poszukujemy regularności (stabilność) oraz staramy się zrozumieć zachowania nieregularne (chaos). Intuicyjnie, nieprzewidywalność wiąże się z pewnym rodzajem "mieszania" w przestrzeni stanów. Celem artykułu jest prezentacja możliwości opisu tego zjawiska w oparciu o pojęcia topologiczne.

EN

When we study processes evolving in time, we usually consider two directions of research: we look for regularity (stability) of dynamics and we try to understand irregular behavior (chaos) which is present in the system. Intuitively, unpredictability of dynamics is related to some kind of "mixing" in the phase space. The aim of this article is to present notions which try to describe this phenomena on a basis of topological approach.

3

Teoria chaosu w ujęciu matematycznym

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Kwietniak D. , Oprocha P.

Matematyka Stosowana : matematyka dla społeczeństwa

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2008

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tom Nr 9

1-45

PL

Niniejsza praca stanowi próbę przedstawienia istniejących definicji chaosu dla dyskretnych układów dynamicznych. Dyskusję zawężono do zagadnień związanych z dynamiką topologiczną. Przedstawiono i umotywowano definicje: wrażliwości na warunki początkowe, chaosu w sensie Li i Yorke’a, Auslandera i Yorke’a, Devaneya, chaosu dystrybucyjnego, entropii topologicznej i podkowy topologicznej. Podzielono się pewnymi uwagami historycznymi. Omówiono znane związki między różnymi definicjami chaosu i przypomniano związane z nimi problemy otwarte.

EN

This work is intended as an attempt to survey existingde finitions of chaos for discrete dynamical systems. Discussion is restricted to the settingof topological dynamics, while the measure-theoretic (ergodic theory) and smooth (differentiable dynamical systems) aspects are omitted as exceedingt he scope of this paper. Chaos theory is understood here as a part of topological dynamics, so aforementioned definitions of chaos are just examples of particular dynamical system properties, and are considered inside the framework of the mathematical theory of discrete dynamical systems. It is not the purpose of this article to study chaos theory understood as a new kind of interdisciplinary branch of science devoted to nonlinear phenomena. As for prerequisites, the reader is expected to possess some mathematical maturity, and to be familiar with basic topology of (compact) metric spaces. No preliminary knowledge of the dynamical systems theory is required, however some is recommended. The first two section are devoted to general discussion of the term "chaos" and contains authors opinion on this subject. To facilitate access to the rest of the article some relevant material from the dynamical system theory is briefly repeated in the third section. The next section (Section 4) introduces the notion of topological transitivity along with some stronger variants, namely topological mixing and weak mixing. Section 5 gives a detailed account of the famous Sharkovskii's Theorem in its full generality. This is required for characterization of chaotic interval maps. Sections 6-13 are devoted to various notions of chaos or related to chaos in dynamical systems. Each section contains an attempt to motivate the notion, historical background and formal definition followed with a review of known properties, relations between various notions of chaos, and some relevant open problems. Section 6 is devoted to a sensitivity to initial conditions – a notion which is accepted as a basic indicator of chaotic behavior. Section 7 introduces a definition of chaos accordingt o Auslander and Yorke. Section 8 examines the notion of Li-Yorke pair and Li-Yorke chaos. Section 9 deals with the definition of chaos introduced in Devaney's book (Devaney chaos). Section 10 recalls some facts connected with symbolic dynamics, which provides a rich source of examples for various interestingb ehavior, and it is an indispensable tool for exploration of many systems. Section 11 describes the so-called "topological horseshoes", which are generalizations of the famous example due to Smale. The existence of a horseshoe in a given dynamical system proves the existence of a subsystem with a dynamics similar to some symbolic dynamical system, hence with a very complicated beTeoria chaosu w ujęciu matematycznym 45 havior. Section 12 gives a brief exposition of the topological entropy and its relation to chaos. The review of various notions of chaos ends with section 13, containingd escription of distributional chaos.

4

A numerical view on topological transitivity and mixing for symbolic dynamics

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Jabłoński D. , Oprocha P.

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2007

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tom Vol. 40, nr 1

229-238

EN

The aim of this paper is to give some algorithms detecting topological transitivity, mixing and other properties of subshifts of finite type.

5

A study of chaos for processes under small perturbations II : rigorous proof of chaos

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Oprocha P. , Wilczyński P.

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tom Vol. 30, no. 1

5-36

EN

In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation [formula]. Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution 0 present in the system when N = 0 are also discussed.

6

Infinite Traces and Symbolic Dynamics - the Minimal Shift Case

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Foryś W. , Oprocha P.

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2011

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tom Vol. 111, nr 2

147-161

EN

We study interrelations between symbolic descriptions of concurrently evolving systems and underlying sequential dynamics. The basic framework for this research is formulated on the background of the theory of traces. We focus our interests on minimal shifts and t-shifts generated by them, that is shifts defined in the space of infinite real traces. We show that sets of infinite real traces generated by minimal shifts are always closed and, under some conditions, are also tshifts. Additional discussion for the case of small alphabets (containing at most four letters) is also provided.

7

Failures prediction based on performance monitoring of a gas turbine: a binary classification approach

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Mulewicz B. , Marzec M. , Morkisz P. , Oprocha P.

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tom Vol. 26

9--21

EN

This paper is dedicated to employ novel technique of deep learning for machines failures prediction. General idea of how to transform sensor data into suitable data set for prediction is presented. Then, neural network architecture that is very successful in solving such problems is derived. Finally, we present a case study for real industrial data of a gas turbine, including results of the experiments.

8

An attempt of optimization of zinc production line

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Jarosz P. , Kusiak J. , Małecki S. , Morkisz P. , Oprocha P. , Pietrucha W. , Sztangret Ł.

Archives of Civil and Mechanical Engineering

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2018

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tom Vol. 18, no. 4

1116--1122

EN

The goal of the research is an attempt of optimization of the hydrometallurgy-based zinc production line, consisting of three stages: mixing of raw materials, oxidative roasting and leaching. The output product of one stage is an input to the next stage. Goal of mixing is preparation of zinc concentrates mix on the basis of zinc concentrates originated from different mines. The output semi-product of the next stage, the oxidative roasting process, is calcine, which is the input of the leaching. The result of the leaching is zinc sulfate solution and the goal of leaching is to carry out the maximum amount of zinc to solution. The preliminary step of any optimization is modeling of the analyzed processes. Modeling of considered three stages of zinc production line, based on the real industrial data of one of zinc production plants, was performed using different techniques. The elaborated models were the basis of the optimization for given objective functions of each of the production stages. The optimization methodology of multi-stage processes developed by the authors was applied. Obtained results of modeling and optimization are presented.

9

Modelowanie procesu produkcji miedzi blister z wykorzystaniem sztucznych sieci neuronowych

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Gulik M. , Jarosz P. , Kusiak J. , Małecki S. , Oprocha P. , Sztangret Ł.

Rudy i Metale Nieżelazne Recykling

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2016

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tom R. 61, nr 1

21--25

PL

Celem artykułu jest analiza możliwości wykorzystania sztucznych sieci neuronowych do modelowania procesu produkcji miedzi blister w piecu zawiesinowym. W szczególności, przedstawiono możliwości modelowania procesu, przy założeniu, że jego parametry nie mogą być zmierzone z wymaganą dokładnością, z czym można się często spotkać w przypadku rzeczywistych procesów przemysłowych. Do budowy modelu wybrano najważniejsze, z technologicznego punktu widzenia, parametry i sporządzono dla nich bilans masy oraz energii. Na jego podstawie stworzono zbiór danych wykorzystany do budowy modelu opartego o sztuczne sieci neuronowe. Opracowany model może być wykorzystany w przyszłości do dalszych badań dotyczących optymalizacji.

EN

The paper presents the analysis of possibilities of application of artificial neural networks to modelling the copper flash smelting process. The Authors focused on ability of the process modelling, assuming that the parameters can’t be measured with a required accuracy. This problem is often found in the case of real industrial processes. The technologically most important parameters were selected.