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Jump Bifurcation and Hysteresis in an Infinite-Dimensional Dynamical System of Coupled Spins*
SIAM J. APPL. MATH. (C) 1990 Society for Industrial and Applied Mathematics Vol. 50, No. 1, pp. 108-124, February 1990 008 JUMP BIFURCATION AND HYSTERESIS IN AN INFINITE-DIMENSIONAL DYNAMICAL SYSTEM OF COUPLED SPINS* RENATO E. MIROLLOt AND STEVEN H. STROGATZ: Abstract. This paper studies an infinite-dimensional dynamical system that models a collection of coupled spins in a random magnetic field. A state of the system is given by a self-map of the unit circle. Critical states for the dynamical system correspond to equilibrium configurations of the spins. Exact solutions are obtained for all the critical states and their stability is characterized by analyzing the second variation of the system's potential function at those states. It is proven rigorously that the system exhibits a jump bifurcation and hysteresis as the coupling between spins is varied. Key words, bifurcation, dynamical system, phase transition, spin system AMS(MOS) subject classifications. 58F14, 82A57, 34F05 1. Introduction. There is currently a great deal of interest in the collective behavior of dynamical systems with many interacting subunits [2]. Such systems are very difficult to analyze rigorously unless some simplifying assumptions are made. "Phase models," in which each subunit is characterized by a single phase angle Oi, provide an analytically tractable class of many-body systems. Phase models have been used in a wide variety of scientific contexts, including charge-density wave transport [8], [9], [23], [24], magnetic spin systems [3], [26], oscillating chemical reactions 15], 18], [28], networks of biological oscillators [4], [7], [14], [27], [28], and arrays of Josephson junctions [11], [25]. -
Lecture 9: Newton Method 9.1 Motivation 9.2 History
10-725: Convex Optimization Fall 2013 Lecture 9: Newton Method Lecturer: Barnabas Poczos/Ryan Tibshirani Scribes: Wen-Sheng Chu, Shu-Hao Yu Note: LaTeX template courtesy of UC Berkeley EECS dept. 9.1 Motivation Newton method is originally developed for finding a root of a function. It is also known as Newton- Raphson method. The problem can be formulated as, given a function f : R ! R, finding the point x? such that f(x?) = 0. Figure 9.1 illustrates another motivation of Newton method. Given a function f, we want to approximate it at point x with a quadratic function fb(x). We move our the next step determined by the optimum of fb. Figure 9.1: Motivation for quadratic approximation of a function. 9.2 History Babylonian people first applied the Newton method to find the square root of a positive number S 2 R+. The problem can be posed as solving the equation f(x) = x2 − S = 0. They realized the solution can be achieved by applying the iterative update rule: 2 1 S f(xk) xk − S xn+1 = (xk + ) = xk − 0 = xk − : (9.1) 2 xk f (xk) 2xk This update rule converges to the square root of S, which turns out to be a special case of Newton method. This could be the first application of Newton method. The starting point affects the convergence of the Babylonian's method for finding the square root. Figure 9.2 shows an example of solving the square root for S = 100. x- and y-axes represent the number of iterations and the variable, respectively. -
Question-And-Answer-Physics-Today
Questions and answers with Steven Strogatz An applied mathematician notes that for many physics topics, including general relativity and climate science, the whole is often not equal to the sum of the parts. By Jeremy N. A. Matthews April 2015 Bookends: Questions and answers with Steven Strogatz Questions and answers with Peter Pesic Questions and answers with Marcelo Gleiser Questions and answers with Amit Hagar Questions and answers with A. Douglas Stone Steven Strogatz is the Jacob Schurman Professor of Applied Mathematics at Cornell University in Ithaca, New York. He conducts research on dynamical systems in physics, biology, and social science. His 1998 Nature article “Collective dynamics of ‘small-world’ networks” is considered a seminal contribution to complexity research. Strogatz is also an active communicator of mathematics and related topics to the public. He frequently appears on public radio and writes for The New Yorker and other publications. A 2015 recipient of the Lewis Thomas Prize for Writing About Science, he has written books that include Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life (Hyperion, 2003) and The Joy of x: A Guided Tour of Math, from One to Infinity (Houghton Mifflin Harcourt, 2012). Strogatz recently published the second edition of a text he wrote more than two decades ago. The new edition of Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 2015) contains new exercises on such topics as protein dynamics and superconductivity and covers applications in such emerging fields as sociophysics and systems biology. Physics Today reviewed the book in the April issue. -
The Newton Fractal's Leonardo Sequence Study with the Google
INTERNATIONAL ELECTRONIC JOURNAL OF MATHEMATICS EDUCATION e-ISSN: 1306-3030. 2020, Vol. 15, No. 2, em0575 OPEN ACCESS https://doi.org/10.29333/iejme/6440 The Newton Fractal’s Leonardo Sequence Study with the Google Colab Francisco Regis Vieira Alves 1*, Renata Passos Machado Vieira 1 1 Federal Institute of Science and Technology of Ceara - IFCE, BRAZIL * CORRESPONDENCE: [email protected] ABSTRACT The work deals with the study of the roots of the characteristic polynomial derived from the Leonardo sequence, using the Newton fractal to perform a root search. Thus, Google Colab is used as a computational tool to facilitate this process. Initially, we conducted a study of the Leonardo sequence, addressing it fundamental recurrence, characteristic polynomial, and its relationship to the Fibonacci sequence. Following this study, the concept of fractal and Newton’s method is approached, so that it can then use the computational tool as a way of visualizing this technique. Finally, a code is developed in the Phyton programming language to generate Newton’s fractal, ending the research with the discussion and visual analysis of this figure. The study of this subject has been developed in the context of teacher education in Brazil. Keywords: Newton’s fractal, Google Colab, Leonardo sequence, visualization INTRODUCTION Interest in the study of homogeneous, linear and recurrent sequences is becoming more and more widespread in the scientific literature. Therefore, the Fibonacci sequence is the best known and studied in works found in the literature, such as Oliveira and Alves (2019), Alves and Catarino (2019). In these works, we can see an investigation and exploration of the Fibonacci sequence and the complexification of its mathematical model and the generalizations that result from it. -
Lecture 1: What Is Calculus?
Math1A:introductiontofunctionsandcalculus OliverKnill, 2012 Lecture 1: What is Calculus? Calculus formalizes the process of taking differences and taking sums. Differences measure change, sums explore how things accumulate. The process of taking differences has a limit called derivative. The process of taking sums will lead to the integral. These two pro- cesses are related in an intimate way. In this first lecture, we look at these two processes in the simplest possible setup, where functions are evaluated only on integers and where we do not take any limits. About 25’000 years ago, numbers were represented by units like 1, 1, 1, 1, 1, 1,... for example carved in the fibula bone of a baboon1. It took thousands of years until numbers were represented with symbols like today 0, 1, 2, 3, 4,.... Using the modern concept of function, we can say f(0) = 0, f(1) = 1, f(2) = 2, f(3) = 3 and mean that the function f assigns to an input like 1001 an output like f(1001) = 1001. Lets call Df(n) = f(n + 1) − f(n) the difference between two function values. We see that the function f satisfies Df(n) = 1 for all n. We can also formalize the summation process. If g(n) = 1 is the function which is constant 1, then Sg(n) = g(0) + g(1) + ... + g(n − 1) = 1 + 1 + ... + 1 = n. We see that Df = g and Sg = f. Now lets start with f(n) = n and apply summation on that function: Sf(n) = f(0) + f(1) + f(2) + .. -
Damien Fair, Never at Rest
Spectrum | Autism Research News https://www.spectrumnews.org PROFILES Rising Star: Damien Fair, never at rest BY SARAH DEWEERDT 9 JANUARY 2019 One day last April, Damien Fair chuckled along with the members of his 30-person lab at their monthly lunch meeting as he attempted to maneuver a bowl of soup, a plate of salad and a pair of crutches all at the same time. The associate professor of behavioral neuroscience at Oregon Health and Science University in Portland had broken his ankle in a city-league basketball game the evening before. Fair waved away offers of help and soon got down to business. “All right, teach us something,” he told the lab members who were presenting their work that day. He mostly let the presenters, a graduate student and a research associate, hold the floor. Fair is tall and trim, with a salt-and- pepper goatee and expressive eyebrows. He laughed appreciatively at the visual puns in their PowerPoint presentation — for example, a photograph of a toddler in a wetsuit, a play on the name ‘FreeSurfer,’ software for analyzing brain scans. “Damien never sweats anything — officially,” says Nico Dosenbach, assistant professor of pediatric neurology at Washington University in St. Louis, Missouri. Underneath his cool exterior, though, Fair is intense and driven, says Dosenbach, who has been Fair’s friend since the two were in graduate school. And Fair is not intimidated by conventional notions of status or rank: He doesn’t hesitate to ask other scientists for their data or time — or to offer his in return. These traits have made Fair one of the most productive and sought-after collaborators in the field of brain imaging. -
Fractal Newton Basins
FRACTAL NEWTON BASINS M. L. SAHARI AND I. DJELLIT Received 3 June 2005; Accepted 14 August 2005 The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton’s method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence. Copyright © 2006 M. L. Sahari and I. Djellit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Isaac Newton discovered what we now call Newton’s method around 1670. Although Newton’s method is an old application of calculus, it was discovered relatively recently that extending it to the complex plane leads to a very interesting fractal pattern. We have focused on complex analysis, that is, studying functions holomorphic on a domain in the complex plane or holomorphic mappings. It is a very exciting field, in which many new phenomena wait to be discovered (and have been discovered). It is very closely linked with fractal geometry, as the basins of attraction for Newton’s method have fractal character in many instances. We are interested in geometric problems, for example, the boundary behavior of conformal mappings. In mathematics, dynamical systems have become popular for another reason: beautiful pictures. -
10-5-19 Workshop Announcement V2
Cornell University Department of Mathematics K-12 Education and Outreach MATH 5080 – Mathematics for Secondary School Teachers October 5, 2019 u 9:00 am – 2:30 pm (lunch provided) u 406 Malott Hall 8:45 – 9:00 am Bagels & Juice (provided) 9:00 – 9:20 am Introductions 9:20 – 10:20 am Business Math for Entrepreneurs Lee Kaltman, M.A.T. (New Roots Charter School) I will describe the course called Business Math for Entrepreneurs, share some student work, and explain how I use real-world experiences to engage and motivate students to want to learn more. 10:30 – 11:30 am The Beauty of Calculus Steven Strogatz, Ph.D. (Cornell University, Mathematics) Based on my new book, Infinite Powers: How Calculus Reveals the Secrets of the Universe, I will present a broad look at the story of calculus, focusing on the connections between calculus and the laws of nature. 11:30 – 12:00 pm Lunch (provided) 12:00 – 12:50 pm Teaching Statistics in the Age of Data Science Michael Nussbaum, Ph.D. (Cornell University, Mathematics) I will try to summarize my 20 years of experience teaching statistics at Cornell, focusing in particular on the basic undergraduate course, which is a slightly more advanced version of AP Statistics. This course always includes a computer component, but new challenges have arisen with the advent of “Data Science,” which promises to revolutionize both the practice and the teaching of statistics, and applied mathematics in general, by total integration with computing. I will conclude with a demonstration of how some of the new CS-inspired courses are taught at Cornell, with on-screen computing using software like R or Python. -
Particle Dynamics in Inertial Microfluidics
PARTICLE DYNAMICS IN INERTIAL MICROFLUIDICS vorgelegt von Master of Science Christian Schaaf ORCID: 0000-0002-4154-5459 an der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Michael Lehmann Erster Gutachter: Prof. Dr. Holger Stark Zweiter Gutachter: Prof. Dr. Roland Netz Tag der wissenschaftlichen Aussprache: 17. September 2020 Berlin 2020 Zusammenfassung Die inertiale Mikrofluidik beschäftigt sich mit laminaren Strömungen von Flüssigkeiten durch mikroskopische Kanäle, bei denen die Trägheitseffekte der Flüssigkeit nicht vernach- lässigt werden können. Befinden sich Teilchen in diesen inertialen Strömungen, ordnen sie sich von selbst an bestimmten Positionen auf der Querschnittsfläche an. Da diese Gleich- gewichtspositionen von den Teilcheneigenschaften abhängen, können so beispielsweise Zellen voneinander getrennt werden. In dieser Arbeit beschäftigen wir uns mit der Dynamik mehrerer fester Teilchen, sowie dem Einfluss der Deformierbarkeit auf die Gleichgewichtsposition einer einzelnen Kapsel. Wir verwenden die Lattice-Boltzmann-Methode, um dieses System zu simulieren. Einen wichtigen Grundstein für das Verständnis mehrerer Teilchen bildet die Dynamik von zwei festen Partikeln. Zunächst klassifizieren wir die möglichen Trajektorien, von denen drei zu ungebundenen Zuständen führen und eine über eine gedämpfte Schwingung in einem gebundenem Zustand endet. Zusätzlich untersuchen wir die inertialen Hubkräfte, welche durch das zweite Teilchen stark beeinflusst werden. Dieser Einfluss hängt vor allem vom Abstand der beiden Teilchen entlang der Flussrichtung ab. Im Anschluss an die Dynamik beschäftigen wir uns genauer mit der Stabilität von Paaren und Zügen bestehend aus mehreren festen Teilchen. Wir konzentrieren uns auf Fälle, in denen die Teilchen sich lateral bereits auf ihren Gleichgewichtspositionen befinden, jedoch nicht entlang der Flussrichtung. -
Top 300 Semifinalists 2016
TOP 300 SEMIFINALISTS 2016 2016 Broadcom MASTERS Semifinalists 2 About Broadcom MASTERS Broadcom MASTERS® (Math, Applied Science, Technology and Engineering for Rising Stars), a program of Society for Science & the Public, is the premier middle school science and engineering fair competition. Society-affiliated science fairs around the country nominate the top 10% of sixth, seventh and eighth grade projects to enter this prestigious competition. After submitting the online application, the top 300 semifinalists are selected. Semifinalists are honored for their work with a prize package that includes an award ribbon, semifinalist certificate of accomplishment, Broadcom MASTERS backpack, a Broadcom MASTERS decal, a one year family digital subscription to Science News magazine, an Inventor's Notebook and copy of “Howtoons” graphic novel courtesy of The Lemelson Foundation, and a one year subscription to Mathematica+, courtesy of Wolfram Research. In recognition of the role that teachers play in the success of their students, each semifinalist's designated teacher also will receive a Broadcom MASTERS tote bag and a one year subscription to Science News magazine, courtesy of KPMG. From the semifinalist group, 30 finalists are selected and will present their research projects and compete in hands-on team STEM challenges to demonstrate their skills in critical thinking, collaboration, communication and creativity at the Broadcom MASTERS finals. Top awards include a grand prize of $25,000, trips to STEM summer camps and more. Broadcom Foundation and Society for Science & the Public thank the following for their support of 2016 Broadcom MASTERS: • Samueli Foundation • Robert Wood Johnson Foundation • Science News for Students • Wolfram Research • Allergan • Affiliated Regional and State Science • Computer History Museum & Engineering Fairs • Deloitte. -
Dynamical Systems with Applications Using Mathematicar
DYNAMICAL SYSTEMS WITH APPLICATIONS USING MATHEMATICA⃝R SECOND EDITION Stephen Lynch Springer International Publishing Contents Preface xi 1ATutorialIntroductiontoMathematica 1 1.1 AQuickTourofMathematica. 2 1.2 TutorialOne:TheBasics(OneHour) . 4 1.3 Tutorial Two: Plots and Differential Equations (One Hour) . 7 1.4 The Manipulate Command and Simple Mathematica Programs . 9 1.5 HintsforProgramming........................ 12 1.6 MathematicaExercises. .. .. .. .. .. .. .. .. .. 13 2Differential Equations 17 2.1 Simple Differential Equations and Applications . 18 2.2 Applications to Chemical Kinetics . 27 2.3 Applications to Electric Circuits . 31 2.4 ExistenceandUniquenessTheorem. 37 2.5 MathematicaCommandsinTextFormat . 40 2.6 Exercises ............................... 41 3PlanarSystems 47 3.1 CanonicalForms ........................... 48 3.2 Eigenvectors Defining Stable and Unstable Manifolds . 53 3.3 Phase Portraits of Linear Systems in the Plane . 56 3.4 Linearization and Hartman’s Theorem . 59 3.5 ConstructingPhasePlaneDiagrams . 61 3.6 MathematicaCommands .. .. .. .. .. .. .. .. .. 69 3.7 Exercises ............................... 70 4InteractingSpecies 75 4.1 CompetingSpecies .......................... 76 4.2 Predator-PreyModels . .. .. .. .. .. .. .. .. .. 79 4.3 Other Characteristics Affecting Interacting Species . 84 4.4 MathematicaCommands .. .. .. .. .. .. .. .. .. 86 4.5 Exercises ............................... 87 v vi CONTENTS 5LimitCycles 91 5.1 HistoricalBackground . .. .. .. .. .. .. .. .. .. 92 5.2 Existence and Uniqueness of -
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.