Appendix a the Mathematics of Discontinuity
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Robust Learning of Chaotic Attractors
Robust Learning of Chaotic Attractors Rembrandt Bakker* Jaap C. Schouten Marc-Olivier Coppens Chemical Reactor Engineering Chemical Reactor Engineering Chemical Reactor Engineering Delft Univ. of Technology Eindhoven Univ. of Technology Delft Univ. of Technology [email protected]·nl [email protected] [email protected]·nl Floris Takens C. Lee Giles Cor M. van den Bleek Dept. Mathematics NEC Research Institute Chemical Reactor Engineering University of Groningen Princeton Nl Delft Univ. of Technology F. [email protected] [email protected] [email protected]·nl Abstract A fundamental problem with the modeling of chaotic time series data is that minimizing short-term prediction errors does not guarantee a match between the reconstructed attractors of model and experiments. We introduce a modeling paradigm that simultaneously learns to short-tenn predict and to locate the outlines of the attractor by a new way of nonlinear principal component analysis. Closed-loop predictions are constrained to stay within these outlines, to prevent divergence from the attractor. Learning is exceptionally fast: parameter estimation for the 1000 sample laser data from the 1991 Santa Fe time series competition took less than a minute on a 166 MHz Pentium PC. 1 Introduction We focus on the following objective: given a set of experimental data and the assumption that it was produced by a deterministic chaotic system, find a set of model equations that will produce a time-series with identical chaotic characteristics, having the same chaotic attractor. The common approach consists oftwo steps: (1) identify a model that makes accurate short tenn predictions; and (2) generate a long time-series with the model and compare the nonlinear-dynamic characteristics of this time-series with the original, measured time-series. -
Chaotic Advection, Diffusion, and Reactions in Open Flows Tamás Tél, György Károlyi, Áron Péntek, István Scheuring, Zoltán Toroczkai, Celso Grebogi, and James Kadtke
Chaotic advection, diffusion, and reactions in open flows Tamás Tél, György Károlyi, Áron Péntek, István Scheuring, Zoltán Toroczkai, Celso Grebogi, and James Kadtke Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 10, 89 (2000); doi: 10.1063/1.166478 View online: http://dx.doi.org/10.1063/1.166478 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/10/1?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.173.125.76 On: Mon, 24 Mar 2014 14:44:01 CHAOS VOLUME 10, NUMBER 1 MARCH 2000 Chaotic advection, diffusion, and reactions in open flows Tama´sTe´l Institute for Theoretical Physics, Eo¨tvo¨s University, P.O. Box 32, H-1518 Budapest, Hungary Gyo¨rgy Ka´rolyi Department of Civil Engineering Mechanics, Technical University of Budapest, Mu¨egyetem rpk. 3, H-1521 Budapest, Hungary A´ ron Pe´ntek Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093-0238 Istva´n Scheuring Department of Plant Taxonomy and Ecology, Research Group of Ecology and Theoretical Biology, Eo¨tvo¨s University, Ludovika te´r 2, H-1083 Budapest, Hungary Zolta´n Toroczkai Department of Physics, University of Maryland, College Park, Maryland 20742-4111 and Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 Celso Grebogi Institute for Plasma Research, University of Maryland, College Park, Maryland 20742 James Kadtke Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093-0238 ͑Received 30 July 1999; accepted for publication 8 November 1999͒ We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. -
Synchronization of Chaotic Systems
Synchronization of chaotic systems Cite as: Chaos 25, 097611 (2015); https://doi.org/10.1063/1.4917383 Submitted: 08 January 2015 . Accepted: 18 March 2015 . Published Online: 16 April 2015 Louis M. Pecora, and Thomas L. Carroll ARTICLES YOU MAY BE INTERESTED IN Nonlinear time-series analysis revisited Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 097610 (2015); https:// doi.org/10.1063/1.4917289 Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 121102 (2017); https:// doi.org/10.1063/1.5010300 Attractor reconstruction by machine learning Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 061104 (2018); https:// doi.org/10.1063/1.5039508 Chaos 25, 097611 (2015); https://doi.org/10.1063/1.4917383 25, 097611 © 2015 Author(s). CHAOS 25, 097611 (2015) Synchronization of chaotic systems Louis M. Pecora and Thomas L. Carroll U.S. Naval Research Laboratory, Washington, District of Columbia 20375, USA (Received 8 January 2015; accepted 18 March 2015; published online 16 April 2015) We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. -
Boenig-Liptsin / Entrepreneurs of the Mind / 1 Citizen's Mind
Margo Boenig-Liptsin SIGCIS Workshop, November 9, 2014 Work in !rogress Session "orking draft – please do not cite (ear SIGCIS Workshop !articipants) *hank you $or reading this partial draft of one of my dissertation 'hapters) I-m submitting to your review a draft of a 'hapter $rom the mi##&e of my dissertation, therefore to contextualize the "riting that you "i&& be reading, I inc&ude be&ow an abstract of my "hole dissertation project and a chapter out&ine. 2s you "i&& see, this early draft of Chapter 2 is based primari&y on empirical research of my three national cases1 I haven't yet integrated into it any secondary &iterature. I "ould be very gratef,& $or your suggestions of re$erences that you $eel "ould be interesting $or me to incorporate/refect upon, especial&y $rom the &iterature on the history of computing, given the topi's that I am exploring in this chapter or in the dissertation as a "hole. *his early draft is also missing complete $ootnotes—they are current&y 0ust notes $or mysel$ and references that I need to tidy up. I-m gratef,& $or al& of your comments, "hich "i&& no doubt make the next #raft of this signifcant&y better) Best "ishes, Margo (issertation abstract Making the Citi/en of the Information 2ge7 2 comparative study of computer &iteracy programs $or 'hi&dren, 1960s-1990s My dissertation is a comparative history of the frst computer &iteracy programs $or chi&dren. *he project examines how programs to introduce 'hi&dren to computers in the 9nited States, :rance, and the Soviet 9nion $rom the 1960s to 1990s embodied political, epistemic, and moral debates about the kind of citi/en required $or &i$e in the 21st century. -
Nikolay Luzin, His Students, Adversaries, and Defenders (Notes
Nikolay Luzin, his students, adversaries, and defenders (notes on the history of Moscow mathematics, 1914-1936) Yury Neretin This is historical-mathematical and historical notes on Moscow mathematics 1914-1936. Nikolay Luzin was a central figure of that time. Pavel Alexandroff, Nina Bari, Alexandr Khinchin, Andrey Kolmogorov, Mikhail Lavrentiev, Lazar Lyusternik, Dmitry Menshov, Petr Novikov, Lev Sсhnirelman, Mikhail Suslin, and Pavel Urysohn were his students. We discuss the time of the great intellectual influence of Luzin (1915-1924), the time of decay of his school (1922-1930), a moment of his administrative power (1934-1936), and his fall in July 1936. But the thing which served as a source of Luzin’s inner drama turned out to be a source of his subsequent fame... Lazar Lyusternik [351] Il est temps que je m’arr ete: voici que je dis, ce que j’ai d´eclar´e, et avec raison, ˆetre inutile `adire. Henri Lebesgue, Preface to Luzin’s book, Leˇcons sur les ensembles analytiques et leurs applications, [288] Прошло сто лет и что ж осталось От сильных, гордых сих мужей, Столь полных волею страстей? Их поколенье миновалось Alexandr Pushkin ’Poltava’ There is a common idea that a life of Nikolay Luzin can be a topic of a Shakespeare drama. I am agree with this sentence but I am extremely far from an intention to realize this idea. The present text is an impassive historical-mathematical and historical investigation of Moscow mathematics of that time. On the other hand, this is more a story of its initiation and turning moments than a history of achievements. -
Nonlinear Dynamics of Duffing System with Fractional Order Damping
Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009 August 30 - September 2, 2009, San Diego, California, USA Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009 August 30 - September 2, 2009, San Diego, California, USA DETC2009-86401 DETC2009-86401 NONLINEAR DYNAMICS OF DUFFING SYSTEM WITH FRACTIONAL ORDER DAMPING Junyi Cao Chengbin Ma Research Institute of Diagnostics and Cybernetics, UM-SJTU Joint Institute, Shanghai Jiaotong School of Mechanical engineering, Xi’an Jiaotong University, Shanghai 200240, China University, Xi’an 710049, China [email protected] [email protected] Hang Xie Zhuangde Jiang Research Institute of Diagnostics and Cybernetics, Institute of Precision Engineering, School of School of Mechanical engineering, Xi’an Jiaotong Mechanical engineering, Xi'an Jiaotong University, University, Xi’an 710049, China Xi'an 710049, China ABSTRACT Although some of the mathematical issues remain unsolved, In this paper, nonlinear dynamics of Duffing system with fractional calculus based modeling of complicated dynamics is fractional order damping is investigated. The four order Runge- becoming a recent focus of interest. The dynamics of fractional Kutta method and ten order CFE-Euler methods are introduced order system equations for Chua, Lorenz, Rossler, Chen, Jerk to simulate the fractional order Duffing equations. The effect of and Duffing are mainly investigated [3-9]. It is obvious that the taking fractional order on the system dynamics is investigated chaotic attractors existing in their fractional systems have using phase diagrams, bifurcation diagrams and Poincare map. different fractional orders. -
Learning from Benoit Mandelbrot
2/28/18, 7(59 AM Page 1 of 1 February 27, 2018 Learning from Benoit Mandelbrot If you've been reading some of my recent posts, you will have noted my, and the Investment Masters belief, that many of the investment theories taught in most business schools are flawed. And they can be dangerous too, the recent Financial Crisis is evidence of as much. One man who, prior to the Financial Crisis, issued a challenge to regulators including the Federal Reserve Chairman Alan Greenspan, to recognise these flaws and develop more realistic risk models was Benoit Mandelbrot. Over the years I've read a lot of investment books, and the name Benoit Mandelbrot comes up from time to time. I recently finished his book 'The (Mis)Behaviour of Markets - A Fractal View of Risk, Ruin and Reward'. So who was Benoit Mandelbrot, why is he famous, and what can we learn from him? Benoit Mandelbrot was a Polish-born mathematician and polymath, a Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) who developed a new branch of mathematics known as 'Fractal' geometry. This geometry recognises the hidden order in the seemingly disordered, the plan in the unplanned, the regular pattern in the irregularity and roughness of nature. It has been successfully applied to the natural sciences helping model weather, study river flows, analyse brainwaves and seismic tremors. A 'fractal' is defined as a rough or fragmented shape that can be split into parts, each of which is at least a close approximation of its original-self. -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Role of Nonlinear Dynamics and Chaos in Applied Sciences
v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings. -
A New Complex Duffing Oscillator Used in Complex Signal Detection
Article Electronics June 2012 Vol.57 No.17: 21852191 doi: 10.1007/s11434-012-5145-8 SPECIAL TOPICS: A new complex Duffing oscillator used in complex signal detection DENG XiaoYing*, LIU HaiBo & LONG Teng School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China Received December 12, 2011; accepted February 2, 2012 In view of the fact that complex signals are often used in the digital processing of certain systems such as digital communication and radar systems, a new complex Duffing equation is proposed. In addition, the dynamical behaviors are analyzed. By calculat- ing the maximal Lyapunov exponent and power spectrum, we prove that the proposed complex differential equation has a chaotic solution or a large-scale periodic one depending on different parameters. Based on the proposed equation, we present a complex chaotic oscillator detection system of the Duffing type. Such a dynamic system is sensitive to the initial conditions and highly immune to complex white Gaussian noise, so it can be used to detect a weak complex signal against a background of strong noise. Results of the Monte-Carlo simulation show that the proposed detection system can effectively detect complex single frequency signals and linear frequency modulation signals with a guaranteed low false alarm rate. complex Duffing equation, chaos detection, weak complex signal, signal-to-noise ratio, probability of detection Citation: Deng X Y, Liu H B, Long T. A new complex Duffing oscillator used in complex signal detection. Chin Sci Bull, 2012, 57: 21852191, doi: 10.1007/s11434-012-5145-8 Chaos theory is one of the most significant achievements of plex chaos control and synchronization, and so on. -
Interactive Effects of Shifting Body Size and Feeding Adaptation Drive
bioRxiv preprint doi: https://doi.org/10.1101/101675; this version posted January 20, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 1 Interactive effects of shifting body size and feeding 2 adaptation drive interaction strengths of protist 3 predators under warming 4 Temperature adaptation of feeding 5 K.E.Fussmann 1,2, B. Rosenbaum 2, 3, U.Brose 2, 3, B.C.Rall 2, 3 6 1 J.F. Blumenbach Institute of Zoology and Anthropology, University of Göttingen, Berliner 7 Str. 28, 37073 Göttingen, Germany 8 2 German Centre for Integrative Biodiversity Research (iDiv), Halle-Jena-Leipzig, Deutscher 9 Platz 5e, 04103 Leipzig, Germany 10 3 Institute of Ecology, Friedrich Schiller University Jena, Dornburger-Str. 159, 07743 Jena, 11 Germany 12 Corresponding author: Katarina E. Fussmann 13 telephone: +49 341 9733195 14 email: [email protected] 15 Keywords: climate change, functional response, body size, temperature adaptation, activation 16 energies, microcosm experiments, predator-prey, interaction strength, Bayesian statistics 17 Paper type: Primary Research 1 bioRxiv preprint doi: https://doi.org/10.1101/101675; this version posted January 20, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 18 Abstract 19 Global change is heating up ecosystems fuelling biodiversity loss and species extinctions. -
Dr. Michael L. Rosenzweig the Man the Scientist the Legend
BIOL 7083 Community Ecologist Presentation Dr. Michael L. Rosenzweig The Man The Scientist The Legend Michael Rosenzweigs Biographical Information Born in 1941 Jewish Parents wanted him to be a physician Ph.D. University of Pennsylvania, 1966 Advisor: Robert H. MacArthur, Ph.D. Married for over 40 years to Carole Ruth Citron Together they have three children, and several grandchildren Biographical Information Known to be an innovator Founded the Department of Ecology & Evolutionary Biology at the University of Arizona in 1975, and was its first head In 1987 he founded the scientific journal Evolutionary Ecology In 1998, when prices for journals began to rise, he founded a competitor, Evolutionary Ecology Research Honor and Awards Ecological Society of America Eminent Ecologist Award for 2008 Faculty of Sci, Univ Arizona, Career Teaching Award, 2001 Ninth Lukacs Symp: Twentieth Century Distinguished Service Award, 1999 International Ecological Soc: Distinguished Statistical Ecologist, 1998 Udall Center for Studies in Public Policy, Univ Arizona: Fellow, 1997±8 Mountain Research Center, Montana State Univ: Distinguished Lecturer, 1997 Univ Umeå, Sweden: Distinguished Visiting Scholar, 1997 Univ Miami: Distinguished Visiting Professor, 1996±7 Univ British Columbia: Dennis Chitty Lecturer, 1995±6 Iowa State Univ: 30th Paul L. Errington Memorial Lecturer, 1994 Michigan State Univ, Kellogg Biological Station: Eminent Ecologist, 1992 Honor and Awards Ben-Gurion Univ of the Negev, Israel: Jacob Blaustein Scholar, 1992