DOCSLIB.ORG

Synthesis of the Advance in and Application of Fractal Characteristics of Traffic Flow

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Synthesis of the Advance in and Application of Fractal Characteristics of Traffic Flow Synthesis of the Advance in and Application of Fractal Characteristics of Traffic Flow Final Report Contract No. BDK80 977‐25 July 2013 Prepared by: Lehman Center for Transportation Research Florida International University Prepared for: Research Center Florida Department of Transportation Final Report Contract No. BDK80 977-25 Synthesis of the Advance in and Application of Fractal Characteristics of Traffic Flow Prepared by: Kirolos Haleem, Ph.D., P.E., Research Associate Priyanka Alluri, Ph.D., Research Associate Albert Gan, Ph.D., Professor Lehman Center for Transportation Research Department of Civil and Environmental Engineering Florida International University 10555 West Flagler Street, EC 3680 Miami, FL 33174 Phone: (305) 348-3116 Fax: (305) 348-2802 and Hongtai Li, Graduate Research Assistant Tao Li, Ph.D., Associate Professor School of Computer Science Florida International University 11200 SW 8th Street Miami, FL 33199 Phone: (305) 348-6036 Fax: (305) 348-3549 Prepared for: Research Center State of Florida Department of Transportation 605 Suwannee Street, M.S. 30 Tallahassee, FL 32399-0450 July 2013 DISCLAIMER The opinions, findings, and conclusions expressed in this publication are those of the authors and not necessarily those of the State of Florida Department of Transportation. iii METRIC CONVERSION CHART SYMBOL WHEN YOU KNOW MULTIPLY BY TO FIND SYMBOL LENGTH in inches 25.4 millimeters mm ft feet 0.305 meters m yd yards 0.914 meters m mi miles 1.61 kilometers km mm millimeters 0.039 inches in m meters 3.28 feet ft m meters 1.09 yards yd km kilometers 0.621 miles mi SYMBOL WHEN YOU KNOW MULTIPLY BY TO FIND SYMBOL AREA in2 square inches 645.2 square millimeters mm2 ft2 square feet 0.093 square meters m2 yd2 square yard 0.836 square meters m2 ac acres 0.405 hectares ha mi2 square miles 2.59 square kilometers km2 mm2 square millimeters 0.0016 square inches in2 m2 square meters 10.764 square feet ft2 m2 square meters 1.195 square yards yd2 ha hectares 2.47 acres ac km2 square kilometers 0.386 square miles mi2 SYMBOL WHEN YOU KNOW MULTIPLY BY TO FIND SYMBOL VOLUME fl oz fluid ounces 29.57 milliliters mL gal gallons 3.785 liters L ft3 cubic feet 0.028 cubic meters m3 yd3 cubic yards 0.765 cubic meters m3 mL milliliters 0.034 fluid ounces fl oz L liters 0.264 gallons gal m3 cubic meters 35.314 cubic feet ft3 m3 cubic meters 1.307 cubic yards yd3 NOTE: volumes greater than 1000 L shall be shown in m3 iv Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. 4. Title and Subtitle 5. Report Date Synthesis of the Advance in and Application of Fractal Characteristics of July 2013 Traffic Flow 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. Kirolos Haleem, Priyanka Alluri, Albert Gan, Hongtai Li, and Tao Li 9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) Lehman Center for Transportation Research Florida International University 11. Contract or Grant No. 10555 West Flagler Street, EC 3680, Miami, FL 33174 BDK80 977-25 12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered Research Center Final Report State of Florida Department of Transportation June 2012 – July 2013 605 Suwannee Street, M.S. 30, Tallahassee, Florida 32399-0450 14. Sponsoring Agency Code 99700-3596-119 15. Supplementary Notes Mr. Jacques Defrant, P.E., of the Florida Department of Transportation (FDOT) District 4 succeeded Mr. Felix Delgado, P.E., formerly also of FDOT District 4, as the Project Manager for this project. 16. Abstract Fractals are irregular geometric objects that exhibit finite details at all scales, and once magnified, their basic structures remain the same regardless of the scale of magnification. Fractal theory has been successfully applied in different fields of science. This project provides a synthesis of existing applications of fractal theory in various fields, as well as its potential applications in traffic management. The specific information gathered and summarized in this report includes: (1) a synthesis of fractal applications in fields that share similarities with transportation networks, such as electrical networks, and in fields outside of transportation; (2) a synthesis of fractal applications that have been proven effective in traffic flow; (3) additional insights on how fractal theory can be applied in traffic management strategies; and (4) summary of research findings and recommendations for more detailed research. Two fractal techniques, the fractal dimension and the Hurst exponent, were applied to detect the existence of fractal characteristics in traffic and crash data from Florida. Traffic volume, speed, and occupancy data obtained from 15- min and 1-hr detector data at two locations in Miami-Dade County were found to exhibit fractal characteristics. Furthermore, the speed trend revealed stronger fractal behavior compared to the volume and occupancy trends for the same time period. The existence of fractal characteristics in crash data was detected in both annual and daily frequency trends. However, the daily crash frequency trends exhibited greater extent of fractal behavior compared to the annual trends, mainly due to the existence of more random fluctuations. The fractal investigation of both annual and three-year average crash rates at ten randomly-selected signalized intersections revealed that the annual crash rate trend, in general, exhibited relatively more fractal characteristics than the three-year average trend. Future research could make use of the insights presented in this study to apply fractal theory in traffic management strategies such as managed lanes, ramp metering, crash analysis, and travel time reliability. Compared to traditional models, fractal theory is anticipated to yield more precise estimates of performance measures. Therefore, a potential future research is to apply fractal theory for predicting short-term traffic flow. Another promising avenue is to apply fractal theory to identify high-crash locations and to predict crash rate at specific locations. These approaches could potentially result in increased efficiency, mobility, and safety of the entire roadway network. 17. Key Word 18. Distribution Statement Fractal Theory, Fractal Characteristics, Traffic Flow, Traffic Prediction, Traffic Management, Traffic Safety 19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price Unclassified Unclassified 91 Form DOT F 1700.7 (8-72) Reproduction of completed page authorized v ACKNOWLEDGEMENTS This research was funded by the Research Center of the Florida Department of Transportation (FDOT) under the direction of Mr. Darryll Dockstader. We are particularly grateful to our Project Manager, Mr. Jacques Defrant, P.E., of the FDOT District 4, for his guidance and support throughout the project. We are also grateful to Mr. Felix Delgado, P.E., formerly also of the FDOT District 4, who preceded Mr. Defrant as the Project Manager. Last but certainly not the least, we would like to thank Ms. Vicki Morrison of the FDOT Research Center for her editing of this report. vi EXECUTIVE SUMMARY The main objective of this project is to provide a synthesis of existing applications of fractal theory in various fields, as well as potential applications of fractal theory in traffic management. The study also identifies those elements of traffic flow which exhibit fractal characteristics. The study further uses crash and traffic flow data from Florida to investigate the existence of fractal characteristics. The specific information gathered and summarized in this report includes: 1. A synthesis of fractal applications in fields that share similarities with transportation networks, such as electrical and water supply networks, and in fields outside of transportation, such as astronomy and ecology; 2. A synthesis of fractal applications that have been proven effective in traffic flow such as in traffic flow prediction; 3. Additional insights on how fractal theory can be applied to traffic management strategies, such as managed lanes, ramp metering, etc.; and 4. Summary of research findings and recommendations for more detailed research and development of future guidance documents. Fractals are irregular geometric objects that exhibit finite details at all scales, and when magnified, their basic structures remain the same regardless of the scale of magnification. The main characteristics of fractals are self-similarity, iterative process, infinite complexity, and existence of non-integer complex dimension. Common fractal analysis techniques are the Hurst exponent, the fractal dimension, the largest Lyapunov exponent, the power spectrum, and the Kolmogorov entropy. The main indicators of existence of fractal characteristics are as follows: the largest Lyapunov exponent should have a positive sign; the Hurst exponent (H) estimate should lie between 0 and 1; and the Kolmogorov entropy should have a positive sign. Moreover, larger estimates of the Lyapunov exponent, the Hurst exponent, and the fractal dimension indicate more complex fractal nonlinear characteristics. Fractal theory has been successfully applied in different fields of science, including but not limited to animal behavior studies, human health studies, economics, astronomy, ecology, physics, pavement engineering, environmental engineering, behavior of materials, and electrical networks. Even though application of fractal theory is not uncommon in several areas in transportation, such
Load more

Recommended publications
  • Pictures of Julia and Mandelbrot Sets
    Pictures of Julia and Mandelbrot Sets Wikibooks.org January 12, 2014 On the 28th of April 2012 the contents of the English as well as German Wikibooks and Wikipedia projects were licensed under Cre- ative Commons Attribution-ShareAlike 3.0 Unported license. An URI to this license is given in the list of figures on page 143. If this document is a derived work from the contents of one of these projects and the content was still licensed by the project under this license at the time of derivation this document has to be licensed under the same, a similar or a compatible license, as stated in section 4b of the license. The list of contributors is included in chapter Contributors on page 141. The licenses GPL, LGPL and GFDL are included in chapter Licenses on page 151, since this book and/or parts of it may or may not be licensed under one or more of these licenses, and thus require inclusion of these licenses. The licenses of the figures are given in the list of figures on page 143. This PDF was generated by the LATEX typesetting software. The LATEX source code is included as an attachment (source.7z.txt) in this PDF file. To extract the source from the PDF file, we recommend the use of http://www.pdflabs.com/tools/pdftk-the-pdf-toolkit/ utility or clicking the paper clip attachment symbol on the lower left of your PDF Viewer, selecting Save Attachment. After ex- tracting it from the PDF file you have to rename it to source.7z.
    [Show full text]
  • 4.3 Discovering Fractal Geometry in CAAD
    4.3 Discovering Fractal Geometry in CAAD Francisco Garcia, Angel Fernandez*, Javier Barrallo* Facultad de Informatica. Universidad de Deusto Bilbao. SPAIN E.T.S. de Arquitectura. Universidad del Pais Vasco. San Sebastian. SPAIN * Fractal geometry provides a powerful tool to explore the world of non-integer dimensions. Very short programs, easily comprehensible, can generate an extensive range of shapes and colors that can help us to understand the world we are living. This shapes are specially interesting in the simulation of plants, mountains, clouds and any kind of landscape, from deserts to rain-forests. The environment design, aleatory or conditioned, is one of the most important contributions of fractal geometry to CAAD. On a small scale, the design of fractal textures makes possible the simulation, in a very concise way, of wood, vegetation, water, minerals and a long list of materials very useful in photorealistic modeling. Introduction Fractal Geometry constitutes today one of the most fertile areas of investigation nowadays. Practically all the branches of scientific knowledge, like biology, mathematics, geology, chemistry, engineering, medicine, etc. have applied fractals to simulate and explain behaviors difficult to understand through traditional methods. Also in the world of computer aided design, fractal sets have shown up with strength, with numerous software applications using design tools based on fractal techniques. These techniques basically allow the effective and realistic reproduction of any kind of forms and textures that appear in nature: trees and plants, rocks and minerals, clouds, water, etc. For modern computer graphics, the access to these techniques, combined with ray tracing allow to create incredible landscapes and effects.
    [Show full text]
  • What Are Lyapunov Exponents, and Why Are They Interesting?
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016.
    [Show full text]
  • Rendering Hypercomplex Fractals Anthony Atella [email protected]
    Rhode Island College Digital Commons @ RIC Honors Projects Overview Honors Projects 2018 Rendering Hypercomplex Fractals Anthony Atella [email protected] Follow this and additional works at: https://digitalcommons.ric.edu/honors_projects Part of the Computer Sciences Commons, and the Other Mathematics Commons Recommended Citation Atella, Anthony, "Rendering Hypercomplex Fractals" (2018). Honors Projects Overview. 136. https://digitalcommons.ric.edu/honors_projects/136 This Honors is brought to you for free and open access by the Honors Projects at Digital Commons @ RIC. It has been accepted for inclusion in Honors Projects Overview by an authorized administrator of Digital Commons @ RIC. For more information, please contact [email protected]. Rendering Hypercomplex Fractals by Anthony Atella An Honors Project Submitted in Partial Fulfillment of the Requirements for Honors in The Department of Mathematics and Computer Science The School of Arts and Sciences Rhode Island College 2018 Abstract Fractal mathematics and geometry are useful for applications in science, engineering, and art, but acquiring the tools to explore and graph fractals can be frustrating. Tools available online have limited fractals, rendering methods, and shaders. They often fail to abstract these concepts in a reusable way. This means that multiple programs and interfaces must be learned and used to fully explore the topic. Chaos is an abstract fractal geometry rendering program created to solve this problem. This application builds off previous work done by myself and others [1] to create an extensible, abstract solution to rendering fractals. This paper covers what fractals are, how they are rendered and colored, implementation, issues that were encountered, and finally planned future improvements.
    [Show full text]
  • 오늘의 천체사진(APOD) 번역집 2013-4 Bigcrunch 소개글
    오늘의 천체사진(APOD) 번역집 2013-4 BigCrunch 소개글 NASA에서 운영하는 오늘의 천체사진(Astronomy Picture of the Day, 이하 APOD)사이트는 1995년 6월 16일 첫번째 천체사진을 발표한 이래, 지금까지 하루도 빠짐없이 천체관련 사진 또는 동영상을 발표하고 있습니다. 본 블로그 북은 2013년 APOD를 통해 발표된 천체사진의 번역 모음집 두번째 편으로서 2013년 11월 1일부터 12월 31일까지 총 51개의 APOD 번역내역이 담겨 있습니다. 참고 : 2013년 발표 천체사진 중 과거에 이미 발표된 내용이 반복 게재된 경우, 그리고 동영상이 게재되어 본 블로그북 형식으로 담아낼 수 없는 경우는 제외되었으니 이점 참고하여 주십시오. 목차 1 목성의 삼중 일식 6 2 하이브리드 일식 9 3 뉴욕 상공의 일식 13 4 노르웨이 상공의 오로라 16 5 1만 3천미터 상공의 일식 20 6 우간다 상공의 일식 23 7 러브조이 혜성과 M44 26 8 불꽃놀이와 번개 사이의 혜성 30 9 개기일식 동안 촬영된 왕성한 활동력의 태양 33 10 토성의 그림자 속에서 바라본 토성 36 11 NGC 1097 39 12 태양의 스펙트럼 42 13 아이손 혜성 45 14 맥너트 혜성의 웅장한 꼬리 48 15 아이슬란드 상공의 오로라와 멋진 구름 51 16 4U1630-47 의 블랙홀 제트 54 17 미노타우르스 미사일의 궤적 57 18 캘리포니아 성운과 플레아데스 성단 60 19 스테레오 위성이 포착한 아이손 혜성 63 20 인디안 코브 상공의 헤일-밥 혜성 66 21 NGC 4921 69 22 시에라 네바다 산맥 위의 모자 구름 72 23 NGC 1999 75 24 태양 주위를 통과하기 전과 후의 아이손 혜성 79 25 은하 중심을 향해 발사되고 있는 레이저 82 26 M63 앞을 지나는 러브조이 혜성 86 27 로 오피유키(Rho Ophiuchi)의 찬란한 구름들 89 28 러브조이 혜성(C/2013 R1) 92 29 Abell 7 95 30 감마선을 내뿜는 지구와 하늘 98 31 구제프 크레이터 전경 - 에베레스트 파노라마(the Everest panorama) 102 32 풍차 위의 러브조이 혜성 105 33 세이퍼트의 6중 은하(Seyfert's Sextet) 108 34 지구에서 가장 추운 곳 111 35 알니탁, 알니렘, 민타카 114 36 다샨바오 습지 상공의 쌍동이 자리 유성우 118 37 NGC 7635 121 38 유로파 124 39 달에 내려선 탐사로봇 위투(Yutu) 127 40 테이드 화산 위의 유성우 130 41 핀란드에서 촬영된 빛기둥 133 42 다채로운 색깔의 달 136 43 호수의 세계 타이탄 140 44 태양활동관측위성이 다파장으로 바라본 태양 144 45 칠레에서 촬영된 쌍동이자리 유성우 148 46 Sharpless 2-308 151 47 M33의 수소구름 154 48 하트 성운 중심의 Melotte 15 159 49 알라스카의 오로라 162 50 환상적인 프렉탈의 풍경 165 51 말머리 성운 169 01 목성의 삼중 일식 목성의 삼중 일식 2013.11.16 22:02 세계표준시 10월 12일 새벽 5시 28분에 벨기에에서 촬영된 이 사진은 천체망원경에 웹카메라를 이용하여 촬영한 것으로서 목성의 달이 목성 상공을 지나면서 그 그림자를 목성 대기에 드리우고 있는 모습을 보여주고 있다.
    [Show full text]
  • Is Type 1 Diabetes a Chaotic Phenomenon?
    Is type 1 diabetes a chaotic phenomenon? Jean-Marc Ginoux, Heikki Ruskeepää, Matjaž Perc, Roomila Naeck, Véronique Di Costanzo, Moez Bouchouicha, Farhat Fnaiech, Mounir Sayadi, Takoua Hamdi To cite this version: Jean-Marc Ginoux, Heikki Ruskeepää, Matjaž Perc, Roomila Naeck, Véronique Di Costanzo, et al.. Is type 1 diabetes a chaotic phenomenon?. Chaos, Solitons and Fractals, Elsevier, 2018, 111, pp.198-205. 10.1016/j.chaos.2018.03.033. hal-02194779 HAL Id: hal-02194779 https://hal.archives-ouvertes.fr/hal-02194779 Submitted on 29 Jul 2019 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. Is type 1 diabetes a chaotic phenomenon? Jean-Marc Ginoux,1, † Heikki Ruskeep¨a¨a,2, ‡ MatjaˇzPerc,3, 4, 5, § Roomila Naeck,6 V´eronique Di Costanzo,7 Moez Bouchouicha,1 Farhat Fnaiech,8 Mounir Sayadi,8 and Takoua Hamdi8 1Laboratoire d’Informatique et des Syst`emes, UMR CNRS 7020, CS 60584, 83041 Toulon Cedex 9, France 2University of Turku, Department of Mathematics and Statistics, FIN-20014 Turku, Finland 3Faculty of Natural Sciences and Mathematics, University of
    [Show full text]
  • Math Morphing Proximate and Evolutionary Mechanisms
    Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales.
    [Show full text]
  • A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks
    entropy Article A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks Bulcsú Sándor 1,* , Bence Schneider 1, Zsolt I. Lázár 1 and Mária Ercsey-Ravasz 1,2,* 1 Department of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania; [email protected] (B.S.); [email protected] (Z.I.L.) 2 Network Science Lab, Transylvanian Institute of Neuroscience, 400157 Cluj-Napoca, Romania * Correspondence: [email protected] (B.S.); [email protected] (M.E.-R.) Abstract: The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for character- izing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos. Keywords: Lyapunov exponents; state-transition networks; time series analysis; dynamical systems Citation: Sándor, B.; Schneider, B.; Lázár, Z.I.; Ercsey-Ravasz, M.
    [Show full text]
  • Emergent Chaos in the Verge of Phase Transitions
    UNIVERSIDAD DE LOS ANDES BACHELOR THESIS Emergent chaos in the verge of phase transitions Author: Supervisor: Jerónimo VALENCIA Dr. Gabriel TÉLLEZ A thesis submitted in fulfillment of the requirements for the degree of Bachelor in Physics in the Department of Physics January 14, 2019 iii Declaration of Authorship I, Jerónimo VALENCIA, declare that this thesis titled, “Emergent chaos in the verge of phase transitions” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research degree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have con- tributed myself. Signed: Date: v “The fluttering of a butterfly’s wing in Rio de Janeiro, amplified by atmospheric currents, could cause a tornado in Texas two weeks later.” Edward Norton Lorenz “We avoid the gravest difficulties when, giving up the attempt to frame hypotheses con- cerning the constitution of matter, we pursue statistical inquiries as a branch of rational mechanics” Josiah Willard Gibbs vii UNIVERSIDAD DE LOS ANDES Abstract Faculty of Sciences Department of Physics Bachelor in Physics Emergent chaos in the verge of phase transitions by Jerónimo VALENCIA The description of phase transitions on different physical systems is usually done using Yang and Lee, 1952, theory.
    [Show full text]
  • Application of Lyapunov Exponents to Strange Attractors and Intact & Damaged Ship Stability
    Application of Lyapunov Exponents to Strange Attractors and Intact & Damaged Ship Stability William R. Story Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science In Ocean Engineering Leigh McCue, Chair Alan Brown Wayne Neu April 29, 2009 Blacksburg, Virginia Tech Keywords: Stability, Capsize, Lyapunov, Attractor, Lorenz Application of Lyapunov Exponents to Strange Attractors and Intact & Damaged Ship Stability William R. Story (ABSTRACT) The threat of capsize in unpredictable seas has been a risk to vessels, sailors, and cargo since the beginning of a seafaring culture. The event is a nonlinear, chaotic phenomenon that is highly sensitive to initial conditions and difficult to repeatedly predict. In extreme sea states most ships depend on an operating envelope, relying on the operator’s detailed knowledge of headings and maneuvers to reduce the risk of capsize. While in some cases this mitigates this risk, the nonlinear nature of the event precludes any certainty of dynamic vessel stability. This research presents the use of Lyapunov exponents, a quantity that measures the rate of trajectory separation in phase space, to predict capsize events for both intact and damaged stability cases. The algorithm searches backwards in ship motion time histories to gather neighboring points for each instant in time, and then calculates the exponent to measure the stretching of nearby orbits. By measuring the periods between exponent maxima, the lead‐ time between period spike and extreme motion event can be calculated. The neighbor‐ searching algorithm is also used to predict these events, and in many cases proves to be the superior method for prediction.
    [Show full text]
  • Fdlyapu: Estimating the Lyapunov Exponents Spectrum Sylvain Mangiarotti & Mireille Huc 2019-08-29
    FDLyapu: Estimating the Lyapunov exponents spectrum Sylvain Mangiarotti & Mireille Huc 2019-08-29 The FDLyapu package is an extention of the GPoM package1. Its aim is to assess the spectrum of the Lyapunov exponents for dynamical systems of polynomial form. The fractal dimension can also be deduced from this spectrum. The algorithms available in the package are based on the algebraic formulation of the equations. The GPoM package, which enables the numerical formulation of algebraic equations in a polynomial form, is thus directly required for this purpose. A user-friendly interface is also provided with FDLyapu, although the codes can be used in a blind mode. The connexion with GPoM is very natural here since this package aims to obtain Ordinary Differential Equations (ODE) from observational time series. The present package can thus be applied afterwards to characterize the global models obtained with GPoM but also to any dynamical system of ODEs expressed in polynomial form. Algorithms The Lyapunov exponents quantify the rate of separation of infinitesimally close trajectories. Depending on the initial conditions, this rate can be positive (divergence) or negative (convergence). A n-dimensionnal system is characterized by n characteristic Lyapunov exponents. For continuous dynamical systems, one Lyapunov exponent must correspond to the direction of the flow and should thus equal zero in average. To estimate the Lyapunov exponents spectrum, two algorithms are made available to the user, both based on the formal derivation of the Jacobian matrix (derivation is actually semi-formal only since the numeric coefficients are used in the derivation process). ¤ Method 1 (Wolf) The first algorithm made available in the package was introduced by Wolf et al.
    [Show full text]
  • Fractal Texture: a Survey
    Advances in Computational Research ISSN: 0975-3273 & E-ISSN: 0975-9085, Volume 5, Issue 1, 2013, pp.-149-152. Available online at http://www.bioinfopublication.org/jouarchive.php?opt=&jouid=BPJ0000187 FRACTAL TEXTURE: A SURVEY RANI M.1 AND AGGARWAL S.2* 1Department of Mathematics, Statistics & Computer Science, Central University of Rajasthan, Ajmer- 305 801, Rajasthan, India. 2Department of MCA, Krishna Engineering College, Mohan Nagar, Ghaziabad- 201 007, UP, India. *Corresponding Author: Email- [email protected] Received: August 07, 2013; Accepted: September 02, 2013 Abstract- The places where you can find fractals include almost every part of the universe, from bacteria cultures to galaxies to your body. Many natural surfaces have a statistical quality of roughness and self-similarity at different scales. Mandelbrot proposed fractal geometry and is the first one to notice its existence in the natural world. Fractals have become popular recently in computer graphics for generating realistic looking textured images. This paper presents a brief introduction to fractals along with its main features and generation techniques. The con- centration is on the fractal texture and the measures to express the texture of fractals. Keywords- fractal dimension, fractal texture, lacunarity, succolarity Citation: Rani M. and Aggarwal S. (2013) Fractal Texture: A Survey. Advances in Computational Research, ISSN: 0975-3273 & E-ISSN: 0975- 9085, Volume 5, Issue 1, pp.-149-152. Copyright: Copyright©2013 Rani M. and Aggarwal S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.
    [Show full text]