Chaos Theory Tamed
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Stability Analysis of Fitzhughâ•Finagumo With
Proceedings of GREAT Day Volume 2011 Article 4 2012 Stability Analysis of FitzHugh–Nagumo with Smooth Periodic Forcing Tyler Massaro SUNY Geneseo Follow this and additional works at: https://knightscholar.geneseo.edu/proceedings-of-great-day Creative Commons Attribution 4.0 License This work is licensed under a Creative Commons Attribution 4.0 License. Recommended Citation Massaro, Tyler (2012) "Stability Analysis of FitzHugh–Nagumo with Smooth Periodic Forcing," Proceedings of GREAT Day: Vol. 2011 , Article 4. Available at: https://knightscholar.geneseo.edu/proceedings-of-great-day/vol2011/iss1/4 This Article is brought to you for free and open access by the GREAT Day at KnightScholar. It has been accepted for inclusion in Proceedings of GREAT Day by an authorized editor of KnightScholar. For more information, please contact [email protected]. Massaro: Stability Analysis of FitzHugh–Nagumo with Smooth Periodic Forcin Stability Analysis of FitzHugh – Nagumo with Smooth Periodic Forcing Tyler Massaro 1 Background Since the concentration of K+ ions is so much higher inside the cell than outside, there is a As Izhikevich so aptly put it, tendency for K+ to flow out of these leak channels “If somebody were to put a gun to the head of along its concentration gradient. When this the author of this book and ask him to name the happens, there is a negative charge left behind by single most important concept in brain science, he the K+ ions immediately leaving the cell. This would say it is the concept of a neuron[16].” build-up of negative charge is actually enough to, in a sense, catch the K+ ions in the act of leaving By no means are the concepts forwarded in his and momentarily halt the flow of charge across the book restricted to brain science. -
Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
At Work in Qualitative Analysis: a Case Study of the Opposite
HISTORIA MATHEMATICA 25 (1998), 379±411 ARTICLE NO. HM982211 The ``Essential Tension'' at Work in Qualitative Analysis: A Case Study of the Opposite Points of View of Poincare and Enriques on the Relationships between Analysis and Geometry* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Giorgio Israel and Marta Menghini Dipartimento di Matematica, UniversitaÁ di Roma ``La Sapienza,'' P. le A. Moro, 2, I00185 Rome, Italy An analysis of the different philosophic and scienti®c visions of Henri Poincare and Federigo Enriques relative to qualitative analysis provides us with a complex and interesting image of the ``essential tension'' between ``tradition'' and ``innovation'' within the history of science. In accordance with his scienti®c paradigm, Poincare viewed qualitative analysis as a means for preserving the nucleus of the classical reductionist program, even though it meant ``bending the rules'' somewhat. To Enriques's mind, qualitative analysis represented the af®rmation of a synthetic, geometrical vision that would supplant the analytical/quantitative conception characteristic of 19th-century mathematics and mathematical physics. Here, we examine the two different answers given at the turn of the century to the question of the relationship between geometry and analysis and between mathematics, on the one hand, and mechanics and physics, on the other. 1998 Academic Press Un'analisi delle diverse posizioni ®loso®che e scienti®che di Henri Poincare e Federigo Enriques nei riguardi dell'analisi qualitativa fornisce un'immagine complessa e interessante della ``tensione essenziale'' tra ``tradizione'' e ``innovazione'' nell'ambito della storia della scienza. -
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 -
Table of Contents More Information
Cambridge University Press 978-1-107-03042-8 - Lyapunov Exponents: A Tool to Explore Complex Dynamics Arkady Pikovsky and Antonio Politi Table of Contents More information Contents Preface page xi 1Introduction 1 1.1 Historical considerations 1 1.1.1 Early results 1 1.1.2 Biography of Aleksandr Lyapunov 3 1.1.3 Lyapunov’s contribution 4 1.1.4 The recent past 5 1.2 Outline of the book 6 1.3 Notations 8 2Thebasics 10 2.1 The mathematical setup 10 2.2 One-dimensional maps 11 2.3 Oseledets theorem 12 2.3.1 Remarks 13 2.3.2 Oseledets splitting 15 2.3.3 “Typical perturbations” and time inversion 16 2.4 Simple examples 17 2.4.1 Stability of fixed points and periodic orbits 17 2.4.2 Stability of independent and driven systems 18 2.5 General properties 18 2.5.1 Deterministic vs. stochastic systems 18 2.5.2 Relationship with instabilities and chaos 19 2.5.3 Invariance 20 2.5.4 Volume contraction 21 2.5.5 Time parametrisation 22 2.5.6 Symmetries and zero Lyapunov exponents 24 2.5.7 Symplectic systems 26 3 Numerical methods 28 3.1 The largest Lyapunov exponent 28 3.2 Full spectrum: QR decomposition 29 3.2.1 Gram-Schmidt orthogonalisation 31 3.2.2 Householder reflections 31 v © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-03042-8 - Lyapunov Exponents: A Tool to Explore Complex Dynamics Arkady Pikovsky and Antonio Politi Table of Contents More information vi Contents 3.3 Continuous methods 33 3.4 Ensemble averages 35 3.5 Numerical errors 36 3.5.1 Orthogonalisation 37 3.5.2 Statistical error 38 3.5.3 -
The Development of Hierarchical Knowledge in Robot Systems
THE DEVELOPMENT OF HIERARCHICAL KNOWLEDGE IN ROBOT SYSTEMS A Dissertation Presented by STEPHEN W. HART Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2009 Computer Science c Copyright by Stephen W. Hart 2009 All Rights Reserved THE DEVELOPMENT OF HIERARCHICAL KNOWLEDGE IN ROBOT SYSTEMS A Dissertation Presented by STEPHEN W. HART Approved as to style and content by: Roderic Grupen, Chair Andrew Barto, Member David Jensen, Member Rachel Keen, Member Andrew Barto, Department Chair Computer Science To R. Daneel Olivaw. ACKNOWLEDGMENTS This dissertation would not have been possible without the help and support of many people. Most of all, I would like to extend my gratitude to Rod Grupen for many years of inspiring work, our discussions, and his guidance. Without his sup- port and vision, I cannot imagine that the journey would have been as enormously enjoyable and rewarding as it turned out to be. I am very excited about what we discovered during my time at UMass, but there is much more to be done. I look forward to what comes next! In addition to providing professional inspiration, Rod was a great person to work with and for|creating a warm and encouraging labora- tory atmosphere, motivating us to stay in shape for his annual half-marathons, and ensuring a sufficient amount of cake at the weekly lab meetings. Thanks for all your support, Rod! I am very grateful to my thesis committee|Andy Barto, David Jensen, and Rachel Keen|for many encouraging and inspirational discussions. -
List of Mathematics, Statistics E Books
Copyright Sr No Book Title Author Year English Package Name OpenURL An Introduction to the Mathematical 1 Theory of the Navier-Stokes Equations Giovanni Galdi 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-09620-9 2 The Proof is in the Pudding Steven G. Krantz 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-48744-1 Inequalities: Theory of Majorization and 3 Its Applications Albert W. Marshall, Ingram Olkin, Barry C. Arnold 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-68276-1 Functional Analysis, Sobolev Spaces and 4 Partial Differential Equations Haim Brezis 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-70914-7 Neutral and Indifference Portfolio Pricing, 5 Hedging and Investing Srdjan Stojanovic 2012 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-71418-9 6 The Real Numbers and Real Analysis Ethan D. Bloch 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-72177-4 7 An Introduction to Hopf Algebras Robert G. Underwood 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-72766-0 8 It's a Nonlinear World Richard H. Enns 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-75340-9 The Colorado Mathematical Olympiad and 9 Further Explorations Alexander Soifer 2011 Mathematics and Statistics http://link.springer.com/openurl?genre=book&isbn=978-0-387-75472-7 -
Remembering Sofia Kovalevskaya
Remembering Sofia Kovalevskaya Michèle Audin Remembering Sofia Kovalevskaya Michèle Audin Institut de recherche mathématique avancée Université de Strasbourg et CNRS 7 rue René-Descartes 67084 Strasbourg Cedex France [email protected] Whilst we have made considerable efforts to contact all holders of copyright material contained in this book. We have failed to locate some of them. Should holders wish to contact the Publisher, we will make every effort to come to some arrangement with them ISBN 978-0-85729-928-4 e-ISBN 978-0-85729-929-1 DOI 10.1007/978-0-85729-929-1 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011935730 Mathematics Subject Classification: 01-00, 01455, 14H70, 70Hxx, 70Exx, 35A10 Translation from the French language edition: ‘Souvenirs sur Sofia Kovalevskaya’ by Michèle Audin Copyright © 2008 Calvage et Mounet, France http://www.calvage-et-mounet.fr/ All Rights Reserved Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copy- right, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. -
Voltage Control Using Limited Communication**This Work Was
20th IFAC World Congress 2017 IFAC PapersOnline Volume 50, Issue 1 Toulouse, France 9-14 July 2017 Part 1 of 24 Editors: Denis Dochain Didier Henrion Dimitri Peaucelle ISBN: 978-1-5108-5071-2 Printed from e-media with permission by: Curran Associates, Inc. 57 Morehouse Lane Red Hook, NY 12571 Some format issues inherent in the e-media version may also appear in this print version. Copyright© (2017) by IFAC (International Federation of Automatic Control) All rights reserved. Printed by Curran Associates, Inc. (2018) For permission requests, please contact the publisher, Elsevier Limited at the address below. Elsevier Limited 360 Park Ave South New York, NY 10010 Additional copies of this publication are available from: Curran Associates, Inc. 57 Morehouse Lane Red Hook, NY 12571 USA Phone: 845-758-0400 Fax: 845-758-2633 Email: [email protected] Web: www.proceedings.com TABLE OF CONTENTS PART 1 VOLTAGE CONTROL USING LIMITED COMMUNICATION*............................................................................................................1 Sindri Magnússon, Carlo Fischione, Na Li A LOCAL STABILITY CONDITION FOR DC GRIDS WITH CONSTANT POWER LOADS* .........................................................7 José Arocas-Pérez, Robert Griño VSC-HVDC SYSTEM ROBUST STABILITY ANALYSIS BASED ON A MODIFIED MIXED SMALL GAIN AND PASSIVITY THEOREM .......................................................................................................................................................................13 Y. Song, C. Breitholtz -
A Russian Childhood Sofya Kovalevskaya a Russian Childhood
A Russian Childhood Sofya Kovalevskaya A Russian Childhood TRANSLATED, EDITED AND INTRODUCED BY BEATRICE STILLMAN With an Analysis of Kovalevskaya' s Mathematics by P. Y. Kochina USSR Academy of Sciences Springer Science+Business Media New York This book has been selected for inclusion in the Sources and Translation Series of the Russian Institute, Columbia University. AMS'classification 01A70 Library of Congress Cataloging in Publication Data Kovalevskafa, Sof; Vasil'evna Korvin-Krukovskafa, 1850-1891. A Russian childhood. Translation of Vospominanifa detstva. Bibliography: p. Includes index. 1. Kovalevskai';, Sofi; Vasil'evna Korvin-Krukovskai-;, 1850- 1891. 2. Mathematicians-Russia-Biography. I. Title. QA29.K67A3513 510'.92'4 [B] 78-12955 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1978 by Beatrice Stillman Originally published by Springer-Verlag New York in 197R. Softcover reprint of the hardcover 1st edition 1978 ISBN 978-1-4419-2808-5 ISBN 978-1-4757-3839-1 (eBook) DOI 10.1007/978-1-4757-3839-1 For Pelageya Y akovlevna Kochina, untiring scholar and generous spirit, in admiration and friendship Contents Pre/ace tX Introduction 1 CHAPTER ONE Earliest Memories 47 CHAPTER TWO The Thief 59 CHAPTER THREE Metamorphosis 77 CHAPTER FOUR Palibino 82 CHAPTER FIVE Mzss Smith 98 vii Contents CHAPTER SIX Uncle Pyotr V asil£evich Krukovsky 111 CHAPTER SEVEN Uncle Fyodor Fyodorovich Shubert 124 CHAPTER EIGHT My Sister 133 CHAPTER NINE Anyuta's Nihilism 145 CHAPTER TEN Anyuta's First Literary Experiments 155 CHAPTER ELEVEN Our Friendship with Fyodor Mikhailovich Dostoevsky 173 Notes 201 An Autobiographical Sketch 213 On the Scientific Work of Sofya Kovalevskaya, by P. -
Arxiv:Hep-Th/0212169V1 15 Dec 2002
LPT–Orsay–02–111 RUB–TP2–17/02 hep-th/0212169 Noncompact Heisenberg spin magnets from high-energy QCD III. Quasiclassical approach ´ 1 2 3 S.E. Derkachov , G.P. Korchemsky and A.N. Manashov ∗ 1 Department of Mathematics, St.-Petersburg Technology Institute, 198013 St.-Petersburg, Russia 2 Laboratoire de Physique Th´eorique†, Universit´ede Paris XI, 91405 Orsay C´edex, France 3 Institut f¨ur Theoretische Physik II, Ruhr–Universit¨at Bochum, 44780 Bochum, Germany Abstract: The exact solution of the noncompact SL(2, C) Heisenberg spin magnet reveals a hidden sym- metry of the energy spectrum. To understand its origin, we solve the spectral problem for arXiv:hep-th/0212169v1 15 Dec 2002 the model within quasiclassical approach. In this approach, the integrals of motion satisfy the Bohr-Sommerfeld quantization conditions imposed on the orbits of classical motion. In the rep- resentation of the separated coordinates, the latter wrap around a Riemann surface defined by the spectral curve of the model. A novel feature of the obtained quantization conditions is that they involve both the α and β periods of the action differential on the Riemann surface, thus allowing us to find their− solutions− by exploring the full modular group of the spectral curve. We demonstrate that the quasiclassical energy spectrum is in a good agreement with the exact results. ∗ Permanent address: Department of Theoretical Physics, Sankt-Petersburg State University, 199034 St.- Petersburg, Russia †Unite Mixte de Recherche du CNRS (UMR 8627) Contents 1. Introduction 2 2. Noncompact Heisenberg spin magnet 6 2.1. Classicalmodel .................................... 6 2.2. -
Unconventional Means of Preventing Chaos in the Economy
CES Working Papers – Volume XIII, Issue 2 Unconventional means of preventing chaos in the economy Gheorghe DONCEAN*, Marilena DONCEAN** Abstract This research explores five barriers, acting as obstacles to chaos in terms of policy (A), economic (B), social (C), demographic effects (D) and natural effects (E) factors for six countries: Russia, Japan, USA, Romania, Brazil and Australia and proposes a mathematical modelling using an original program with Matlab mathematical functions for such systems. The initial state of the systems is then presented, the element that generates the mathematical equation specific to chaos, the block connection diagram and the simulation elements. The research methodology focused on the use of research methods such as mathematical modelling, estimation, scientific abstraction. Starting from the qualitative assessment of the state of a system through its components and for different economic partners, the state of chaos is explained by means of a structure with variable components whose values are equivalent to the economic indices researched. Keywords: chaos, system, Chiua simulator, multiscroll, barriers Introduction In the past four decades we have witnessed the birth, development and maturing of a new theory that has revolutionised our way of thinking about natural phenomena. Known as chaos theory, it quickly found applications in almost every area of technology and economics. The emergence of the theory in the 1960s was favoured by at least two factors. First, the development of the computing power of electronic computers enabled (numerical) solutions to most of the equations that described the dynamic behaviour of certain physical systems of interest, equations that could not be solved using the analytical methods available at the time.