DOCSLIB.ORG

Emergent Chaos in the Verge of Phase Transitions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. In short, it describes a phase as a re- gion of the space of configurations in which certain quantities, state functions, behave in a similar manner and where they vary smoothly. In this work, an attempt of discovering similar characteristics in quantities describing the dy- namical system which arise from Hamilton equations was done. The Lyapunov exponents, Hausdorff dimension and Kolmogorov Sinai entropy gave interest- ing results. In fact, the macroscopical description of the system using statistical mechanics and the microscopical characterization as a dynamical system seems to coincide and this leads to interesting questions of how this relation could be exploited to diagnose critical behavior in a many-body system. ix Acknowledgements The first person to thank for the confidence and unconditional support in my whole life must be my mother. Although some mixed feelings come into play as my aunt has also play an important role in my academical development. In fact my family’s support has always been fundamental throughout this twenty one years, and they all shall be mentioned in a way or another. But my mother and aunt need special mentions. They taught me how to pursuit a dream and do whatever I like doing, and be confident in how everything, eventually, turns out with a good result when it is done with joy and good will. For all patience I want to thank them, and ask them to continue to be an cornerstone in my life, and to be in many occasions a reason to stop, take a deep breath and retake everything with a whole new perspective. I will be always grateful with them for all these years of training to face life responsibly. Regarding academical experiences I would like to thank my friends from Math and Physics department. People whom I have had the pleasure to meet from school and life, and those I met in my first semester in Universidad de los Andes deserve to be mentioned for all laughter and good times spent together. In addition, there are some professors which helped me become a better student and whose guidance I appreciate as it helped me to find an interesting topic to develop this work. In particular I want to acknowledge to my advisor for his time and tips which made this work better. If you are reading these acknowledgements, maybe you have already been listed before, or you correspond to those people that need not be mentioned to know this work is also their fault. Whether it is this case or not, I hope this work is as delectable for you to read as it was for me to write. And most importantly, I expect you to find at least one interesting idea to think beyond phase transitions, chaos theory or their possible relationships. xi Contents Declaration of Authorship iii Abstract vii Acknowledgements ix 1 Chaos Theory1 1.1 Generalities on dynamical systems..................1 1.1.1 Stability and asymptotic behavior..............3 1.2 Chaotic dynamics............................5 1.2.1 Lyapunov exponents......................7 Calculation of principal Lyapunov exponents.......7 Calculation of higher dimensional exponents........8 1.2.2 Dimension............................9 1.2.3 Kolmogorov-Sinai entropy.................. 11 1.3 Examples................................. 11 1.3.1 Lorenz System......................... 11 Trajectories for the Lorenz system.............. 13 Lyapunov exponents...................... 15 Dimension............................ 19 1.3.2 Double Pendulum....................... 19 Trajectories for the double pendulum............ 21 Lyapunov exponents for the double pendulum....... 22 2 Statistical Mechanics 29 2.1 Phase Transitions............................ 29 2.2 Fully Coupled Rotators Model.................... 30 2.2.1 Canonical Partition Function................. 30 2.2.2 Free Energy........................... 32 2.2.3 Magnetization.......................... 34 Ferromagnetic and paramagnetic behavior......... 35 2.2.4 Internal Energy......................... 36 2.3 Numerical simulations for the Fully Coupled Rotators Model.. 37 2.3.1 Magnetization.......................... 38 2.3.2 Fluctuations........................... 39 2.4 Chaos in the Fully Coupled Rotators Model............. 40 xii 2.4.1 Lyapunov exponents...................... 40 2.4.2 Dimension............................ 45 2.4.3 Kolmogorov-Sinai entropy.................. 46 2.5 Conclusions, conjectures and further work............. 51 Bibliography 53 xiii List of Figures 1.1 10 time steps for the canonical Hénon’s 2D Map with initial con- dition (0.8, 0.5) ..............................2 1.2 Numerical solution of equation 1.3 using a fourth-order Runge- Kutta algorithm. An w-limit set and an attractor are present in this dynamical system..........................4 1.3 Lorenz system trajectory for r = 10.0................. 13 1.4 Lorenz system trajectory for r = 18.0................. 14 1.5 Lorenz system trajectory for r = 26.0................. 14 1.6 Lorenz system trajectory for r = 100.0................ 15 1.7 Convergence of Lyapunov exponents calculations for different ini- tial conditions............................... 16 1.8 Convergence of Lyapunov exponents calculations for different ba- sis...................................... 17 1.9 Lyapunov exponents for the Lorenz system as a function of r ... 18 1.10 Dimension of the Lorenz system using Mori’s formula 1.12 as a function of the order parameter of the system............ 19 1.11 Dimension of the Lorenz system using Kaplan-Yorke’s formula 1.13 as a function of the order parameter of the system....... 20 1.12 Double pendulum coordinates and important variables for the system................................... 20 = 1.13 Trajectories for the double pendulum with pq2 0.1........ 23 = 1.14 Trajectories for the double pendulum with pq2 1.0........ 24 = 1.15 Trajectories for the double pendulum with pq2 2.7........ 25 = 1.16 Trajectories for the double pendulum with pq2 3.5........ 26 = 1.17 Trajectories for the double pendulum with pq2 5.0........ 27 1.18 Lyapunov exponents for the double pendulum as a function of pq2 28 2.1 Graphical interpretation for the self-consistency equation for the Fully Coupled Rotators Model..................... 33 2.2 Evolution of the rotators in the paramagnetic regime e = −5.... 35 2.3 Evolution of the rotators in the ferromagnetic regime e = 5.... 36 2.4 Energy evolution using the fourth order Runge-Kutta algorithm for different time steps.......................... 38 2.5 Fluctuations of the Fully Coupled Rotators Model as a function of internal energy.............................. 41 xiv 2.6 Lyapunov exponents of the Fully Coupled Rotators System as a function of energy for different number of particles......... 42 2.7 Value of L characterizing Lyapunov exponents for energies be- L yond the critical value which behave as l1 ∼ (U − Uc) , varying the number of particles of the system................. 43 2.8 Correspondence between critical energies for different values of e and the emergent discontinuity in Lyapunov exponents curves for 400 particles in the system...................... 44 2.9 Hausdorff dimension per degree of freedom estimations for the Fully Coupled Rotators Model as a function of internal energy U for different number of particles.................... 47 2.10 Hausdorff dimension per degree of freedom estimations for the Fully Coupled Rotators Model as a function of the re-scaled en- ergy U/e for varying values of the coupling e............ 48 2.11 Kolmogorov-Sinai entropy per particle for the Fully Coupled Ro- tators Model for varying number of particles............. 49 2.12 Kolmogorov-Sinai entropy per particle for the Fully Coupled Ro- tators Model as a function of U/e for different values of
Load more

Recommended publications
  • Hamiltonian Chaos
    Hamiltonian Chaos Niraj Srivastava, Charles Kaufman, and Gerhard M¨uller Department of Physics, University of Rhode Island, Kingston, RI 02881-0817. Cartesian coordinates, generalized coordinates, canonical coordinates, and, if you can solve the problem, action-angle coordinates. That is not a sentence, but it is classical mechanics in a nutshell. You did mechanics in Cartesian coordinates in introductory physics, probably learned generalized coordinates in your junior year, went on to graduate school to hear about canonical coordinates, and were shown how to solve a Hamiltonian problem by finding the action-angle coordinates. Perhaps you saw the action-angle coordinates exhibited for the harmonic oscillator, and were left with the impression that you (or somebody) could find them for any problem. Well, you now do not have to feel badly if you cannot find them. They probably do not exist! Laplace said, standing on Newton’s shoulders, “Tell me the force and where we are, and I will predict the future!” That claim translates into an important theorem about differential equations—the uniqueness of solutions for given ini- tial conditions. It turned out to be an elusive claim, but it was not until more than 150 years after Laplace that this elusiveness was fully appreciated. In fact, we are still in the process of learning to concede that the proven existence of a solution does not guarantee that we can actually determine that solution. In other words, deterministic time evolution does not guarantee pre- dictability. Deterministic unpredictability or deterministic randomness is the essence of chaos. Mechanical systems whose equations of motion show symp- toms of this disease are termed nonintegrable.
    [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]
  • Annotated List of References Tobias Keip, I7801986 Presentation Method: Poster
    Personal Inquiry – Annotated list of references Tobias Keip, i7801986 Presentation Method: Poster Poster Section 1: What is Chaos? In this section I am introducing the topic. I am describing different types of chaos and how individual perception affects our sense for chaos or chaotic systems. I am also going to define the terminology. I support my ideas with a lot of examples, like chaos in our daily life, then I am going to do a transition to simple mathematical chaotic systems. Larry Bradley. (2010). Chaos and Fractals. Available: www.stsci.edu/~lbradley/seminar/. Last accessed 13 May 2010. This website delivered me with a very good introduction into the topic as there are a lot of books and interesting web-pages in the “References”-Sektion. Gleick, James. Chaos: Making a New Science. Penguin Books, 1987. The book gave me a very general introduction into the topic. Harald Lesch. (2003-2007). alpha-Centauri . Available: www.br-online.de/br- alpha/alpha-centauri/alpha-centauri-harald-lesch-videothek-ID1207836664586.xml. Last accessed 13. May 2010. A web-page with German video-documentations delivered a lot of vivid examples about chaos for my poster. Poster Section 2: Laplace's Demon and the Butterfly Effect In this part I describe the idea of the so called Laplace's Demon and the theory of cause-and-effect chains. I work with a lot of examples, especially the famous weather forecast example. Also too I introduce the mathematical concept of a dynamic system. Jeremy S. Heyl (August 11, 2008). The Double Pendulum Fractal. British Columbia, Canada.
    [Show full text]
  • Lecture Notes on Classical Mechanics for Physics 106Ab Sunil Golwala
    Lecture Notes on Classical Mechanics for Physics 106ab Sunil Golwala Revision Date: September 25, 2006 Introduction These notes were written during the Fall, 2004, and Winter, 2005, terms. They are indeed lecture notes – I literally lecture from these notes. They combine material from Hand and Finch (mostly), Thornton, and Goldstein, but cover the material in a different order than any one of these texts and deviate from them widely in some places and less so in others. The reader will no doubt ask the question I asked myself many times while writing these notes: why bother? There are a large number of mechanics textbooks available all covering this very standard material, complete with worked examples and end-of-chapter problems. I can only defend myself by saying that all teachers understand their material in a slightly different way and it is very difficult to teach from someone else’s point of view – it’s like walking in shoes that are two sizes wrong. It is inevitable that every teacher will want to present some of the material in a way that differs from the available texts. These notes simply put my particular presentation down on the page for your reference. These notes are not a substitute for a proper textbook; I have not provided nearly as many examples or illustrations, and have provided no exercises. They are a supplement. I suggest you skim them in parallel while reading one of the recommended texts for the course, focusing your attention on places where these notes deviate from the texts. ii Contents 1 Elementary Mechanics 1 1.1 Newtonian Mechanics ..................................
    [Show full text]
  • Instructional Experiments on Nonlinear Dynamics & Chaos (And
    Bibliography of instructional experiments on nonlinear dynamics and chaos Page 1 of 20 Colorado Virtual Campus of Physics Mechanics & Nonlinear Dynamics Cluster Nonlinear Dynamics & Chaos Lab Instructional Experiments on Nonlinear Dynamics & Chaos (and some related theory papers) overviews of nonlinear & chaotic dynamics prototypical nonlinear equations and their simulation analysis of data from chaotic systems control of chaos fractals solitons chaos in Hamiltonian/nondissipative systems & Lagrangian chaos in fluid flow quantum chaos nonlinear oscillators, vibrations & strings chaotic electronic circuits coupled systems, mode interaction & synchronization bouncing ball, dripping faucet, kicked rotor & other discrete interval dynamics nonlinear dynamics of the pendulum inverted pendulum swinging Atwood's machine pumping a swing parametric instability instabilities, bifurcations & catastrophes chemical and biological oscillators & reaction/diffusions systems other pattern forming systems & self-organized criticality miscellaneous nonlinear & chaotic systems -overviews of nonlinear & chaotic dynamics To top? Briggs, K. (1987), "Simple experiments in chaotic dynamics," Am. J. Phys. 55 (12), 1083-9. Hilborn, R. C. (2004), "Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics," Am. J. Phys. 72 (4), 425-7. Hilborn, R. C. and N. B. Tufillaro (1997), "Resource Letter: ND-1: nonlinear dynamics," Am. J. Phys. 65 (9), 822-34. Laws, P. W. (2004), "A unit on oscillations, determinism and chaos for introductory physics students," Am. J. Phys. 72 (4), 446-52. Sungar, N., J. P. Sharpe, M. J. Moelter, N. Fleishon, K. Morrison, J. McDill, and R. Schoonover (2001), "A laboratory-based nonlinear dynamics course for science and engineering students," Am. J. Phys. 69 (5), 591-7. http://carbon.cudenver.edu/~rtagg/CVCP/Ctr_dynamics/Lab_nonlinear_dyn/Bibex_nonline..
    [Show full text]
  • Chaos Theory and Robert Wilson: a Critical Analysis Of
    CHAOS THEORY AND ROBERT WILSON: A CRITICAL ANALYSIS OF WILSON’S VISUAL ARTS AND THEATRICAL PERFORMANCES A dissertation presented to the faculty of the College of Fine Arts Of Ohio University In partial fulfillment Of the requirements for the degree Doctor of Philosophy Shahida Manzoor June 2003 © 2003 Shahida Manzoor All Rights Reserved This dissertation entitled CHAOS THEORY AND ROBERT WILSON: A CRITICAL ANALYSIS OF WILSON’S VISUAL ARTS AND THEATRICAL PERFORMANCES By Shahida Manzoor has been approved for for the School of Interdisciplinary Arts and the College of Fine Arts by Charles S. Buchanan Assistant Professor, School of Interdisciplinary Arts Raymond Tymas-Jones Dean, College of Fine Arts Manzoor, Shahida, Ph.D. June 2003. School of Interdisciplinary Arts Chaos Theory and Robert Wilson: A Critical Analysis of Wilson’s Visual Arts and Theatrical Performances (239) Director of Dissertation: Charles S. Buchanan This dissertation explores the formal elements of Robert Wilson’s art, with a focus on two in particular: time and space, through the methodology of Chaos Theory. Although this theory is widely practiced by physicists and mathematicians, it can be utilized with other disciplines, in this case visual arts and theater. By unfolding the complex layering of space and time in Wilson’s art, it is possible to see the hidden reality behind these artifacts. The study reveals that by applying this scientific method to the visual arts and theater, one can best understand the nonlinear and fragmented forms of Wilson's art. Moreover, the study demonstrates that time and space are Wilson's primary structuring tools and are bound together in a self-renewing process.
    [Show full text]
  • Moon-Earth-Sun: the Oldest Three-Body Problem
    Moon-Earth-Sun: The oldest three-body problem Martin C. Gutzwiller IBM Research Center, Yorktown Heights, New York 10598 The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler’s laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today’s many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem.
    [Show full text]
  • Chapter 8 Nonlinear Systems
    Chapter 8 Nonlinear systems 8.1 Linearization, critical points, and equilibria Note: 1 lecture, §6.1–§6.2 in [EP], §9.2–§9.3 in [BD] Except for a few brief detours in chapter 1, we considered mostly linear equations. Linear equations suffice in many applications, but in reality most phenomena require nonlinear equations. Nonlinear equations, however, are notoriously more difficult to understand than linear ones, and many strange new phenomena appear when we allow our equations to be nonlinear. Not to worry, we did not waste all this time studying linear equations. Nonlinear equations can often be approximated by linear ones if we only need a solution “locally,” for example, only for a short period of time, or only for certain parameters. Understanding linear equations can also give us qualitative understanding about a more general nonlinear problem. The idea is similar to what you did in calculus in trying to approximate a function by a line with the right slope. In § 2.4 we looked at the pendulum of length L. The goal was to solve for the angle θ(t) as a function of the time t. The equation for the setup is the nonlinear equation L g θ�� + sinθ=0. θ L Instead of solving this equation, we solved the rather easier linear equation g θ�� + θ=0. L While the solution to the linear equation is not exactly what we were looking for, it is rather close to the original, as long as the angleθ is small and the time period involved is short. You might ask: Why don’t we just solve the nonlinear problem? Well, it might be very difficult, impractical, or impossible to solve analytically, depending on the equation in question.
    [Show 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
    [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]
  • 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]
  • 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]