Determination of Lyapunov Exponents in Discrete Chaotic Models
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Spatial Accessibility to Amenities in Fractal and Non Fractal Urban Patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser
Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser To cite this version: Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser. Spatial accessibility to amenities in fractal and non fractal urban patterns. Environment and Planning B: Planning and Design, SAGE Publications, 2012, 39 (5), pp.801-819. 10.1068/b37132. hal-00804263 HAL Id: hal-00804263 https://hal.archives-ouvertes.fr/hal-00804263 Submitted on 14 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. TANNIER C., VUIDEL G., HOUOT H., FRANKHAUSER P. (2012), Spatial accessibility to amenities in fractal and non fractal urban patterns, Environment and Planning B: Planning and Design, vol. 39, n°5, pp. 801-819. EPB 137-132: Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile TANNIER* ([email protected]) - corresponding author Gilles VUIDEL* ([email protected]) Hélène HOUOT* ([email protected]) Pierre FRANKHAUSER* ([email protected]) * ThéMA, CNRS - University of Franche-Comté 32 rue Mégevand F-25 030 Besançon Cedex, France Tel: +33 381 66 54 81 Fax: +33 381 66 53 55 1 Spatial accessibility to amenities in fractal and non fractal urban patterns Abstract One of the challenges of urban planning and design is to come up with an optimal urban form that meets all of the environmental, social and economic expectations of sustainable urban development. -
An Extended Correlation Dimension of Complex Networks
entropy Article An Extended Correlation Dimension of Complex Networks Sheng Zhang 1,*, Wenxiang Lan 1,*, Weikai Dai 1, Feng Wu 1 and Caisen Chen 2 1 School of Information Engineering, Nanchang Hangkong University, 696 Fenghe South Avenue, Nanchang 330063, China; [email protected] (W.D.); [email protected] (F.W.) 2 Military Exercise and Training Center, Academy of Army Armored Force, Beijing 100072, China; [email protected] * Correspondence: [email protected] (S.Z.); [email protected] (W.L.) Abstract: Fractal and self-similarity are important characteristics of complex networks. The correla- tion dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimen- sion is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects. Keywords: fractal property; correlation dimension; weighted networks; small-world network Citation: Zhang, S.; Lan, W.; Dai, W.; Wu, F.; Chen, C. An Extended 1. Introduction Correlation Dimension of Complex Recently, revealing and characterizing complex systems from a complex networks Networks. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Equivalent Relation Between Normalized Spatial Entropy and Fractal Dimension
Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension Yanguang Chen (Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, P.R. China. E-mail: [email protected]) Abstract: Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy to fractal dimension is not yet clear in both theory and practice. This paper is devoted to revealing the new equivalence relation between spatial entropy and fractal dimension using functional box-counting method. Based on varied regular fractals, the numerical relationship between spatial entropy and fractal dimension is examined. The results show that the ratio of actual entropy (Mq) to the maximum entropy (Mmax) equals the ratio of actual dimension (Dq) to the maximum dimension (Dmax). The spatial entropy and fractal dimension of complex spatial systems can be converted into one another by means of functional box-counting method. The theoretical inference is verified by observational data of urban form. A conclusion is that normalized spatial entropy is equal to normalized fractal dimension. Fractal dimensions proved to be the characteristic values of entropies. In empirical studies, if the linear size of spatial measurement is small enough, a normalized entropy value is infinitely approximate to the corresponding normalized fractal dimension value. Based on the theoretical result, new spatial measurements of urban space filling can be defined, and multifractal parameters can be generalized to describe both simple systems and complex systems. Key words: spatial entropy; multifractals; functional box-counting method; space filling; urban form; Chinese cities 2 1. -
Correlation Dimension for Self-Similar Cantor Sets with Overlaps
FUNDAMENTA MATHEMATICAE 155 (1998) Correlation dimension for self-similar Cantor sets with overlaps by K´arolyS i m o n (Miskolc) and Boris S o l o m y a k (Seattle, Wash.) Abstract. We consider self-similar Cantor sets Λ ⊂ R which are either homogeneous and Λ − Λ is an interval, or not homogeneous but having thickness greater than one. We have a natural labeling of the points of Λ which comes from its construction. In case of overlaps among the cylinders of Λ, there are some “bad” pairs (τ, ω) of labels such that τ and ω label the same point of Λ. We express how much the correlation dimension of Λ is smaller than the similarity dimension in terms of the size of the set of “bad” pairs of labels. 1. Introduction. In the literature there are some results (see [Fa1], [PS] or [Si] for a survey) which show that for a family of Cantor sets of overlapping construction on R, the dimension (Hausdorff or box counting) is almost surely equal to the similarity dimension, that is, the overlap between the cylinders typically does not lead to dimension drop. However, we do not understand the cause of the decrease of dimension in the exceptional cases. In this paper we consider self-similar Cantor sets Λ on R for which either the thickness of Λ is greater than one or Λ is homogeneous and Λ − Λ is an interval. We prove that the only reason for the drop of the correlation dimension is the size of the set of those pairs of symbolic sequences which label the same point of the Cantor set. -
![Arxiv:2010.07103V2 [Physics.Data-An] 4 Dec 2020 Often Shows Superior Performance](https://docslib.b-cdn.net/cover/4436/arxiv-2010-07103v2-physics-data-an-4-dec-2020-often-shows-superior-performance-1674436.webp)
Arxiv:2010.07103V2 [Physics.Data-An] 4 Dec 2020 Often Shows Superior Performance
Breaking Symmetries of the Reservoir Equations in Echo State Networks Joschka Herteux1, a) and Christoph R¨ath1, b) Institut f¨urMaterialphysik im Weltraum, Deutsches Zentrum f¨urLuft- und Raumfahrt, M¨unchnerStr. 20, 82234 Wessling, Germany (Dated: 7 December 2020) Reservoir computing has repeatedly been shown to be extremely successful in the prediction of nonlinear time-series. However, there is no complete understanding of the proper design of a reservoir yet. We find that the simplest popular setup has a harmful symmetry, which leads to the prediction of what we call mirror- attractor. We prove this analytically. Similar problems can arise in a general context, and we use them to explain the success or failure of some designs. The symmetry is a direct consequence of the hyperbolic tangent activation function. Further, four ways to break the symmetry are compared numerically: A bias in the output, a shift in the input, a quadratic term in the readout, and a mixture of even and odd activation functions. Firstly, we test their susceptibility to the mirror-attractor. Secondly, we evaluate their performance on the task of predicting Lorenz data with the mean shifted to zero. The short-time prediction is measured with the forecast horizon while the largest Lyapunov exponent and the correlation dimension are used to represent the climate. Finally, the same analysis is repeated on a combined dataset of the Lorenz attractor and the Halvorsen attractor, which we designed to reveal potential problems with symmetry. We find that all methods except the output bias are able to fully break the symmetry with input shift and quadratic readout performing the best overall. -
Paul S. Addison
Page 1 Chapter 1— Introduction 1.1— Introduction The twin subjects of fractal geometry and chaotic dynamics have been behind an enormous change in the way scientists and engineers perceive, and subsequently model, the world in which we live. Chemists, biologists, physicists, physiologists, geologists, economists, and engineers (mechanical, electrical, chemical, civil, aeronautical etc) have all used methods developed in both fractal geometry and chaotic dynamics to explain a multitude of diverse physical phenomena: from trees to turbulence, cities to cracks, music to moon craters, measles epidemics, and much more. Many of the ideas within fractal geometry and chaotic dynamics have been in existence for a long time, however, it took the arrival of the computer, with its capacity to accurately and quickly carry out large repetitive calculations, to provide the tool necessary for the in-depth exploration of these subject areas. In recent years, the explosion of interest in fractals and chaos has essentially ridden on the back of advances in computer development. The objective of this book is to provide an elementary introduction to both fractal geometry and chaotic dynamics. The book is split into approximately two halves: the first—chapters 2–4—deals with fractal geometry and its applications, while the second—chapters 5–7—deals with chaotic dynamics. Many of the methods developed in the first half of the book, where we cover fractal geometry, will be used in the characterization (and comprehension) of the chaotic dynamical systems encountered in the second half of the book. In the rest of this chapter brief introductions to fractal geometry and chaotic dynamics are given, providing an insight to the topics covered in subsequent chapters of the book. -
Ontologia De Domínio Fractal
MÉTODOS COMPUTACIONAIS PARA A CONSTRUÇÃO DA ONTOLOGIA DE DOMÍNIO FRACTAL Ivo Wolff Gersberg Dissertação de Mestrado apresentada ao Programa de Pós-graduação em Engenharia Civil, COPPE, da Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Mestre em Engenharia Civil. Orientadores: Nelson Francisco Favilla Ebecken Luiz Bevilacqua Rio de Janeiro Agosto de 2011 MÉTODOS COMPUTACIONAIS PARA CONSTRUÇÃO DA ONTOLOGIA DE DOMÍNIO FRACTAL Ivo Wolff Gersberg DISSERTAÇÃO SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE MESTRE EM CIÊNCIAS EM ENGENHARIA CIVIL. Examinada por: ________________________________________________ Prof. Nelson Francisco Favilla Ebecken, D.Sc. ________________________________________________ Prof. Luiz Bevilacqua, Ph.D. ________________________________________________ Prof. Marta Lima de Queirós Mattoso, D.Sc. ________________________________________________ Prof. Fernanda Araújo Baião, D.Sc. RIO DE JANEIRO, RJ - BRASIL AGOSTO DE 2011 Gersberg, Ivo Wolff Métodos computacionais para a construção da Ontologia de Domínio Fractal/ Ivo Wolff Gersberg. – Rio de Janeiro: UFRJ/COPPE, 2011. XIII, 144 p.: il.; 29,7 cm. Orientador: Nelson Francisco Favilla Ebecken Luiz Bevilacqua Dissertação (mestrado) – UFRJ/ COPPE/ Programa de Engenharia Civil, 2011. Referências Bibliográficas: p. 130-133. 1. Ontologias. 2. Mineração de Textos. 3. Fractal. 4. Metodologia para Construção de Ontologias de Domínio. I. Ebecken, Nelson Francisco Favilla et al . II. Universidade Federal do Rio de Janeiro, COPPE, Programa de Engenharia Civil. III. Titulo. iii À minha mãe e meu pai, Basia e Jayme Gersberg. iv AGRADECIMENTOS Agradeço aos meus orientadores, professores Nelson Ebecken e Luiz Bevilacqua, pelo incentivo e paciência. -
Correlation Dimension Analysis of Complex Hydrological Systems: What Information Can the Method Provide?
Correlation Dimension analysis of complex hydrological systems: what information can the method provide? Dissertation zur Erlangung des Doktorgrades des Fachbereichs Geowissenschaften der Freien Universität Berlin vorgelegt von Miaomiao Ma Berlin 2013 1. Gutachter: PD. Dr. Christoph Merz Freie Universität Berlin Fachbereich Geowissenschaften Institut für Geologische Wissenschaften and Leibniz-Zentrum für Agrarlandschaftsforschung (ZALF) Institut für Landschaftswasserhaushalt 2. Gutachter: Prof. Dr. Michael Schneider Freie Universität Berlin Fachbereich Geowissenschaften Institut für Geologische Wissenschaften 3. Gutachter: Prof. Dr. Gunnar Lischeid Universität Potsdam Institut für Erd- und Umweltwissenschaften and Leibniz-Zentrum für Agrarlandschaftsforschung (ZALF) Institut für Landschaftswasserhaushalt Tag der Disputation: 27.06.2013 Preface The present thesis was conducted as my Ph.D. research project “Equilibrating data and model complexity” in the framework of a cooperation between Freie Universität Berlin (FUB) and the China Scholarship Council (CSC). The work outlined in this thesis was carried out over a period of three years (September 2010 to August 2013). Financial support was provided by the CSC. The main chapters of this thesis have been submitted for publications1, to which I contributed as the first author. I have been in charge of the development and conduction of the research work described. I have been responsible for the scientific content and for the preparation of manuscripts, including literature review, data interpretation, preparation of figures and tables, and manuscript writing. Co-authors have played an advisory and supervisory role. The thesis contains the contributions by Björn Thomas for collecting and extracting the meteorological and catchment properties data of 63 catchments in the Federal State of Brandenburg, Germany. 1 Ma M, Lischeid G, Merz C, Thomas B., under review. -
Uniform Lower Bounds on the Dimension of Bernoulli Convolutions
Uniform lower bounds on the dimension of Bernoulli convolutions V. Kleptsyn, M. Pollicott and P. Vytnova∗ Abstract In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure µλ is the probability measure corresponding to the law of the random variable 1 k ξ = ξkλ ; k=0 X where ξk are i.i.d. random variables assuming values 1 and 1 with equal probability 1 − and 2 < λ < 1. In particular, for Bernoulli convolutions we give a uniform lower bound 1 dimH (µλ) 0:96399 for all < λ < 1. ≥ 2 Contents 1 Introduction1 2 Dimension of a measure 10 3 Computing lower bounds 13 4 Proof of Theorem 3 25 5 Diffusion operator (2) 26 Dα,λ 6 Diffusion operator (1) 34 Dα,λ A Numerical data 41 arXiv:2102.07714v2 [math.DS] 28 Feb 2021 1 Introduction In this note we will study stationary measures for certain types of iterated function schemes. We begin with an important example. ∗The first author is partly supported by ANR Gromeov (ANR-19-CE40-0007), by Centre Henri Lebesgue (ANR-11-LABX-0020-01) and by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant ag. No. 075-15-2019-1931. The second author is partly supported by ERC-Advanced Grant 833802-Resonances and EPSRC grant EP/T001674/1. The third author is partly supported by EPSRC grant EP/T001674/1. -
Measurement of Heart Rate Variability Using Correlation Dimension and Entropy
New Jersey Institute of Technology Digital Commons @ NJIT Theses Electronic Theses and Dissertations Winter 1-31-1994 Measurement of heart rate variability using correlation dimension and entropy Su Zhang New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/theses Part of the Electrical and Electronics Commons Recommended Citation Zhang, Su, "Measurement of heart rate variability using correlation dimension and entropy" (1994). Theses. 1702. https://digitalcommons.njit.edu/theses/1702 This Thesis is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Theses by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, -
A Review on the Nonlinear Dynamical System Analysis of Electrocardiogram Signal
Hindawi Journal of Healthcare Engineering Volume 2018, Article ID 6920420, 19 pages https://doi.org/10.1155/2018/6920420 Review Article A Review on the Nonlinear Dynamical System Analysis of Electrocardiogram Signal 1 2 3 4 1 Suraj K. Nayak , Arindam Bit , Anilesh Dey , Biswajit Mohapatra, and Kunal Pal 1Department of Biotechnology and Medical Engineering, National Institute of Technology, Rourkela, Odisha 769008, India 2Department of Biomedical Engineering, National Institute of Technology, Raipur, Chhattisgarh 492010, India 3Department of Electronics and Communication Engineering, Kaziranga University, Jorhat, Assam 785006, India 4Vesaj Patel Hospital, Rourkela, Odisha 769004, India Correspondence should be addressed to Kunal Pal; [email protected] Received 17 October 2017; Revised 13 January 2018; Accepted 27 February 2018; Published 2 May 2018 Academic Editor: Maria Lindén Copyright © 2018 Suraj K. Nayak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Electrocardiogram (ECG) signal analysis has received special attention of the researchers in the recent past because of its ability to divulge crucial information about the electrophysiology of the heart and the autonomic nervous system activity in a noninvasive manner. Analysis of the ECG signals has been explored using both linear and nonlinear methods. However, the nonlinear methods of ECG signal analysis are gaining popularity because of their robustness in feature extraction and classification. The current study presents a review of the nonlinear signal analysis methods, namely, reconstructed phase space analysis, Lyapunov exponents, correlation dimension, detrended fluctuation analysis (DFA), recurrence plot, Poincaré plot, approximate entropy, and sample entropy along with their recent applications in the ECG signal analysis.