Birth of a New Class of Period-Doubling Scaling Behavior As a Result of Bifurcation in the Renormalization Equation
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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. -
Dm/Dt)Vs 2 Fρ = (Duc/Dρ)N = (Mvdv/Dρ)N + (1/2)(Dm/Dρ)V N Y Análogamente La Fuerza Total Sobre M Ahora Será
0 JUAN RIUS-CAMPS LA IRREVERSIBILIDAD Y EL CAOS 8 DE DICIEMBRE DE 2009 EDICIONES ORDIS 1 2 EDICIONES ORDIS GRAN VÍA DE CARLOS III 59 . 2º . 4ª 08028 BARCELONA JUAN RIUS – CAMPS 8 de Diciembre de 2009 3 4 LA IRREVERSIBILIDAD Y EL CAOS ÍNDICE CAPÍTULO I p. 9 LOS FUNDAMENTOS COSMOLÓGICOS DE LA MECÁNICA Y LAS LEYES FUNDAMENTALES DE LA DINÁMICA INTRODUCCIÓN p. 9 MATERIA Y FORMA p. 10 CAPÍTULO II p. 33 DINÁMICA DE SISTEMAS MACÁNICOS IRREVERSIBLES INTRODUCCIÓN p. 33 SISTEMA DE PUNTOS MATERIALES p. 34 A. FUNDAMENTOS p. 34 B. ESTUDIO ANALÍTICO DE LA EXPRESIÓN DE LA FUERZA EN LA NUEVA DINÁMICA (ND) p. 37 C ENTROPÍA: SEGUNDA LEY FUNDAMENTAL p. 44 D SENTIDO CINEMÁTICO DE LA VELOCIDAD ANGULAR ω* p. 50 5 CAPÍTULO III p. 59 ECUACIONES DE ONDA Y “ECUACIONES DE MAXWELL” A. DETERMINACIÓN DE LA ECUACIÓN DE ONDA CUANDO Up = Up(P, t) p. 59 B. DEDUCCIÓN DE LAS “ECUACIONES DE MAXWELL” p. 67 CAPÍTULO IV p. 74 NOTAS COMPLEMENTARIAS. A. CONDICIÓN DE FUERZA CENTRAL (ND) p. 74 B. CONDICIÓN DE FUERZA CENTRAL (Caso singular de espiral logarítmica) p. 75 C. FUERZA CENTRAL (Expresada en coordenadas polares) p.78 81 CAPÍTULO V p. 81 IRREVERSIBILIDAD Y CAOS CAPÍTULO VI p. 110 PRUEBAS EXPERIMENTALES 6 7 8 CAPÍTULO I LOS FUNDAMENTOS COSMOLÓGICOS DE LA MECÁNICA Y LAS LEYES FUNDAMENTALES DE LA DINÁMICA INTRODUCCIÓN Para expresar nuestro punto de partida se hace preciso afirmar que éste no ha sido mayormente físico sino filosófico, en el sentido de la Metafísica de la Naturaleza o Cosmología, es decir, en la vía abierta desde hace siglos por ARISTÓTELES, continuada por TOMÁS de AQUINO y otros pensadores. -
Universality in Transitions to Chaos
. Chaos: Classical and Quantum I: Deterministic Chaos Predrag Cvitanovic´ – Roberto Artuso – Ronnie Mainieri – Gregor Tanner – Gabor´ Vattay —————————————————————- ChaosBook.org version15.8, Oct 18 2016 printed November 10, 2016 ChaosBook.org comments to: [email protected] Chapter 31 Universality in transitions to chaos When you come to a fork in the road, take it! —Yogi Berra he developments that we shall describe next are one of those pleasing de- monstrations of the unity of physics. The key discovery was made by a T physicist not trained to work on problems of turbulence. In the fall of 1975 Mitchell J. Feigenbaum, an elementary particle theorist, discovered a universal transition to chaos in 1-dimensional unimodal map dynamics. At the time the physical implications of the discovery were nil. During the next few years, howe- ver, numerical and mathematical studies established this universality in a number of realistic models in various physical settings, and in 1980 the universality theory passed its first experimental test. The discovery was that large classes of nonlinear systems exhibit transitions to chaos which are universal and quantitatively measurable. This advance was akin to (and inspired by) earlier advances in the theory of phase transitions; for the first time one could, predict and measure ‘critical exponents’ for turbulence. But the breakthrough consisted not so much in discovering a new set of universal numbers, as in developing a new way to solve strongly nonlinear physical pro- blems. Traditionally, we use regular motions (harmonic oscillators, plane waves, free particles, etc.) as zeroth-order approximations to physical systems, and ac- count for weak nonlinearities perturbational. -
Chapter 5: Fractals II Goals: • to Understand Why the Scaling
Chapter 5: Fractals II In this chapter we address the question of why and have the values that they do and why they are the same for many dierent mapping functions. We will write an equation that embodies the observation of dilation symmetry, and Goals: we will nd that this equation determines the values of the exponents and . To understand why the scaling exhibited by period-doubling bi- In other words, knowing that dilation symmetry exists is enough information furcation sequences of one-dimensional maps is universal, and to to determine what the magnication factors must be. This result explains the calculate the scaling exponents. observation of universality (dierent maps having the same exponents). The To examine other natural processes which give rise to fractal book by Cvitanovic in the reference list is a good resource for further reading structures. about this universality. Several arguments in this chapter are based on this source. This chapter continues our study of dilation-symmetric structures, or frac- We rst review how the iteration of a smooth function leads to n-cycles of tals. These structures are sometimes called “self-similar”. First, we will pursue ever longer length. Any smooth, positive function f(x)=rfo(x) that vanishes our investigation of the period-doubling bifurcation sequence of the logistic map for x = 0 and x = 1 must cross the line f = x when the “amplitude” r exceeds and get some understanding of why the scaling exponents and which char- some positive value. When r is just at this threshold value, f 0 = 1, and as r acterize it are universal. -
A Complete Bibliography of the Journal of Statistical Physics: 1980–1989
A Complete Bibliography of the Journal of Statistical Physics: 1980{1989 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 01 March 2019 Version 1.05 Title word cross-reference (1 + 1) [Cor87]. (1=2; 1=2) [Mar87]. (2 + 1) [Cor88]. (d + 1) [Sch86a]. 1 − µjjxjjz [de 88]. 1=2 [TS81, KLS88a]. 1=f [MS83b, NT81]. 1=rn [KH82]. 1=jx − yj2 [ACCN88]. 100 ≤ Γ ≤ 160 [HMV84]. 2n [Kai82]. 3 [BDS86, DC89, Zha89a]. 4 3 4 [Lin87]. 8 [Lin87]. [HP85, deV85]. [HP85, Tou89]. 2 [BR80, BWdF88]. 3 [BWdF88, CR86]. z [BWdF88]. A [Rue86a, Vai89]. AgRBf−g [Spo83]. An (1) 2 [KY88b]. An [KY88b]. B6 [Kra82b]. B7 [Kra82b]. β [GGP86]. β < 8π [LSM89]. β ≥ 2/δ [New87]. d [BL85, O'C84a, OB84, dSVT87, YHM80]. d =2 (1) [GS85]. d = 2 + 1 [BP89]. d =2+ [SH82]. d = 3 [ST81b]. Dn [KY88b]. Dn 4 [KY88b]. δ [OP89]. dv =2∆− γ [ST81b]. g0: φ :d [BK81]. γ [MIH86, SD86]. Γ = 2 [CFS83, FJS83]. H [AC84, BS89b, Del84b, Mar81, Pia87]. i(π/2)kT [Wu86]. K [GGM86, MT84]. L1 [Cer88, Pol89b]. L1 [Ark83]. λ(rφ)4 [GKT84]. ≤ 2 [GM88b, GM96]. M [de 88, KST89]. Zk−1 [May85]. µ [EC81]. N [RG86a, AZ85, Bau83, Bur86, FTZ84, HR83a, Kal81, KD82a, KPS89, KST89, Kur83b, Lav82b, Sch87c, WS82]. n−2 [Sim81a]. O(n) 1 2 4 4 [ABCT86, Dup87, Lar86]. -
Universality in Transitions to Chaos
Chapter 26 Universality in transitions to chaos The developments that we shall describe next are one of those pleasing demonstrations of the unity of physics. The key discovery was made by a physicist not trained to work on problems of turbulence. In the fall of 1975 Mitchell J. Feigenbaum, an elementary particle theorist, discovered a universal transition to chaos in one-dimensional unimodal map dynamics. At the time the physical implications of the discovery were nil. During the next few years, however, numerical and mathematical studies established this universality in a number of realistic models in various physical settings, and in 1980 the universality theory passed its first experimental test. The discovery was that large classes of nonlinear systems exhibit transi- tions to chaos which are universal and quantitatively measurable. This ad- vance was akin to (and inspired by) earlier advances in the theory of phase transitions; for the first time one could, predict and measure “critical ex- ponents” for turbulence. But the breakthrough consisted not so much in discovering a new set of universal numbers, as in developing a new way to solve strongly nonlinear physical problems. Traditionally, we use regular motions (harmonic oscillators, plane waves, free particles, etc.) as zeroth- order approximations to physical systems, and account for weak nonlinear- ities perturbatively. We think of a dynamical system as a smooth system whose evolution we can follow by integrating a set of differential equations. The universality theory tells us that the zeroth-order approximations to strongly nonlinear systems should be quite different. They show an amaz- ingly rich structure which is not at all apparent in their formulation in terms of differential equations; instead, they exhibit self-similar structures which can be encoded by universality equations of a type which we will describe here.