Dmitri Kartofelev, PhD. Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of Solid Mechanics.
|
|
|
- Adelia Patrick
- 5 years ago
- Views:
Transcription
1 Lecture 15: Fractals and fractal geometry, coastline paradox, spectral characteristics of dynamical systems, 1-D complex valued maps, Mandelbrot set and nonlinear dynamical systems, introduction to application of fractal geometry and chaos Dmitri Kartofelev, PhD Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of Solid Mechanics Week 15 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 1 / 34
2 Lecture outline (discussed during the lecture) Fractal similarity dimension and the coastline paradox Definition of fractal Spectral characteristics of periodic, quasi-periodic and chaotic systems Second look at 1-D and 2-D maps, complex valued maps Mandelbrot set and Julia sets, connection to nonlinear dynamical systems Generation of Mandelbrot set and corresponding Julia sets Buddhabrot Multibrot sets Examples of fractal geometry in nature and applications Introduction to application of fractals and chaos Synchronisation D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 2 / 34
3 Coastline paradox Example of fractal geometry in practical applications: The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded by Benoit Mandelbrot. Read: B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, New Series, 156(3775), 1967, pp See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 3 / 34
4 Coastline paradox x = b x = a Slope of the log-log plot d = ln(l( x)) ln( x) = ln(l( x)) ln(1/ x), (1) where L is resulting measurement and x is measurement resolution (accuracy). D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 4 / 34
5 Coastline paradox D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 5 / 34
6 Coastline paradox, Estonia Resulting length = 809 km See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 6 / 34
7 Coastline paradox, Estonia Resulting length = 1473 km See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 7 / 34
8 Coastline paradox, Kock snowflake Measured length Measured length See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 8 / 34
9 Coastline paradox Great Britain d = 1.25; Norway d = 1.52; Estonia d = 1.2; South African coast d = 1.0 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 15 9 / 34
10 Definition of fractal Fractal Endless and complex geometric shape with fine structure at arbitrarily small scales. In other words magnification of tiny features of fractal are reminiscent of the whole. Similarity can be exact (invariant), more often it is approximate or statistical. Examples: Cantor set, von Kock curve, Hilbert curve, L-systems, etc. See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
11 Spectral characteristics of dynamical systems Spectral Power Level Power spectrum of x(t), periodic Frequency ω 0 Spectral Power Level Power spectrum of sin(ω(t)), quasi-periodic Power spectrum of x(t), chaotic Frequency ω Spectral Power Level Frequency ω D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
12 Dynamics analysis methods Discrete time analysis Orbit diagram long term behaviour. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
13 1-D complex maps, non-trivial dynamics Fixing the polynomial System dynamics D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
14 Mandelbrot set and dynamical systems Mandelbrot set M is defined as follows z n+1 = zn 2 + c, {z, c} C, n Z + z 0 = 0 c M lim sup z n+1 2 n (2) Im(c) Re(c) See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
15 Mandelbrot set, self-similar properties (video) No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
16 Julia sets and dynamical systems Julia set J is defined as follows z n+1 = zn 2 + c, {z, c} C, n Z + c = const = c 2 z 0 J lim sup z n+1 2 n (3) 0.5 Im(z) Re(z) Figure: Filled Julia set where c = i. See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
17 Mandelbrot set and filled Julia sets D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
18 Mandelbrot set and Julia sets D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
19 Fractal dimension of the edge D = 2.0 D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
20 Mandelbrot set and period-p cycles D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
21 Mandelbrot set, Buddhabrot D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
22 Multibrot sets z n+1 = z p n + c, z 0 = 0 (4) Im(c) 0 0 Im(c) 0 0 Im(c) Re(c) Re(c) Re(c) p = 3 p = 4 p = 5 See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
23 Multibrot sets z n+1 = z p n + c, z 0 = Im(c) 0 0 Im(c) 0 0 Im(c) Re(c) Re(c) Re(c) p = 5.5 p = 10 p = 45 See Mathematica.nb file uploaded to course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
24 Fractal geometry of nature D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
25 Fractal geometry of nature D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
26 Fractal geometry of nature D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
27 Fractal geometry of nature (computer graphics) D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
28 Fractal geometry of nature (computer graphics) Brownian noise with d = 2.0 topography. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
29 Fractal geometry and technology D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
30 Synchronisation: metronomes No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
31 Synchronisation: fireflies No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
32 Synchronisation: Millennium bridge (2000) No embedded video files in this version D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
33 Conclusions Fractal similarity dimension and the coastline paradox Definition of fractal Spectral characteristics of periodic, quasi-periodic and chaotic systems Second look at 1-D and 2-D maps, complex valued maps Mandelbrot set and Julia sets, connection to nonlinear dynamical systems Generation of Mandelbrot set and corresponding Julia sets Buddhabrot Multibrot sets Examples of fractal geometry in nature and applications Introduction to application of fractals and chaos Synchronisation D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
34 Revision questions Explain the coastline paradox. Can a coastline be described with Euclidean geometry? What determines spectral characteristics of dynamical systems? What is 1-D complex valued map? What are Mandelbrot set and Julia sets? Physical meaning of the Mandelbrot set? Physical meaning of Julia sets? What is the Multibrot set? Name an example of self-similar phenomena in nature. D. Kartofelev Nonlinear Dynamics EMR 0060 Week / 34
Dmitri Kartofelev, PhD. Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of Solid Mechanics
Lecture 14: Fractals and fractal geometry, coastline paradox, spectral characteristics of dynamical systems, 1-D complex valued maps, Mandelbrot set and nonlinear dynamical systems, introduction to application
Fractals. Justin Stevens. Lecture 12. Justin Stevens Fractals (Lecture 12) 1 / 14
Fractals Lecture 12 Justin Stevens Justin Stevens Fractals (Lecture 12) 1 / 14 Outline 1 Fractals Koch Snowflake Hausdorff Dimension Sierpinski Triangle Mandelbrot Set Justin Stevens Fractals (Lecture
Lecture 1: Introduction, history, dynamics, nonlinearity, 1-D problem, phase portrait
Lecture 1: Introduction, history, dynamics, nonlinearity, 1-D problem, phase portrait Dmitri Kartofelev, PhD Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of
Mathematical Modelling Lecture 14 Fractals
Lecture 14 phil.hasnip@york.ac.uk Overview of Course Model construction dimensional analysis Experimental input fitting Finding a best answer optimisation Tools for constructing and manipulating models
Patterns in Nature 8 Fractals. Stephan Matthiesen
Patterns in Nature 8 Fractals Stephan Matthiesen How long is the Coast of Britain? CIA Factbook (2005): 12,429 km http://en.wikipedia.org/wiki/lewis_fry_richardson How long is the Coast of Britain? 12*200
Fractals, Dynamical Systems and Chaos. MATH225 - Field 2008
Fractals, Dynamical Systems and Chaos MATH225 - Field 2008 Outline Introduction Fractals Dynamical Systems and Chaos Conclusions Introduction When beauty is abstracted then ugliness is implied. When good
Chapter 8. Fractals. 8.1 Introduction
Chapter 8 Fractals 8.1 Introduction Nested patterns with some degree of self-similarity are not only found in decorative arts, but in many natural patterns as well (see Figure 8.1). In mathematics, nested
Zoology of Fatou sets
Math 207 - Spring 17 - François Monard 1 Lecture 20 - Introduction to complex dynamics - 3/3: Mandelbrot and friends Outline: Recall critical points and behavior of functions nearby. Motivate the proof
Fractal Geometry Time Escape Algorithms and Fractal Dimension
NAVY Research Group Department of Computer Science Faculty of Electrical Engineering and Computer Science VŠB- TUO 17. listopadu 15 708 33 Ostrava- Poruba Czech Republic Basics of Modern Computer Science
INTRODUCTION TO FRACTAL GEOMETRY
Every mathematical theory, however abstract, is inspired by some idea coming in our mind from the observation of nature, and has some application to our world, even if very unexpected ones and lying centuries
Fractals and Dimension
Chapter 7 Fractals and Dimension Dimension We say that a smooth curve has dimension 1, a plane has dimension 2 and so on, but it is not so obvious at first what dimension we should ascribe to the Sierpinski
CMSC 425: Lecture 12 Procedural Generation: Fractals
: Lecture 12 Procedural Generation: Fractals Reading: This material comes from classic computer-graphics books. Procedural Generation: One of the important issues in game development is how to generate
The importance of beings fractal
The importance of beings fractal Prof A. J. Roberts School of Mathematical Sciences University of Adelaide mailto:anthony.roberts@adelaide.edu.au May 4, 2012 Good science is the ability to look at things
An Investigation of Fractals and Fractal Dimension. Student: Ian Friesen Advisor: Dr. Andrew J. Dean
An Investigation of Fractals and Fractal Dimension Student: Ian Friesen Advisor: Dr. Andrew J. Dean April 10, 2018 Contents 1 Introduction 2 1.1 Fractals in Nature............................. 2 1.2 Mathematically
Take a line segment of length one unit and divide it into N equal old length. Take a square (dimension 2) of area one square unit and divide
Fractal Geometr A Fractal is a geometric object whose dimension is fractional Most fractals are self similar, that is when an small part of a fractal is magnified the result resembles the original fractal
Dynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0
UNIVERSITY OF TORONTO Faculty of Arts and Science DECEMBER 2011 EXAMINATIONS. MAT335H1F Solutions Chaos, Fractals and Dynamics Examiner: D.
General Comments: UNIVERSITY OF TORONTO Faculty of Arts and Science DECEMBER 2011 EXAMINATIONS MAT335H1F Solutions Chaos, Fractals and Dynamics Examiner: D. Burbulla Duration - 3 hours Examination Aids:
Julia Sets and the Mandelbrot Set
December 14, 2007 : Dynamical System s Example Dynamical System In general, a dynamical system is a rule or function that describes a point s position in space over time, where time can be modeled as a
Intermittency, Fractals, and β-model
Intermittency, Fractals, and β-model Lecture by Prof. P. H. Diamond, note by Rongjie Hong I. INTRODUCTION An essential assumption of Kolmogorov 1941 theory is that eddies of any generation are space filling
18.175: Lecture 2 Extension theorems, random variables, distributions
18.175: Lecture 2 Extension theorems, random variables, distributions Scott Sheffield MIT Outline Extension theorems Characterizing measures on R d Random variables Outline Extension theorems Characterizing
Chaos & Recursive. Ehsan Tahami. (Properties, Dynamics, and Applications ) PHD student of biomedical engineering
Chaos & Recursive Equations (Properties, Dynamics, and Applications ) Ehsan Tahami PHD student of biomedical engineering Tahami@mshdiau.a.ir Index What is Chaos theory? History of Chaos Introduction of
CMSC 425: Lecture 11 Procedural Generation: Fractals and L-Systems
CMSC 425: Lecture 11 Procedural Generation: ractals and L-Systems Reading: The material on fractals comes from classic computer-graphics books. The material on L-Systems comes from Chapter 1 of The Algorithmic
biologically-inspired computing lecture 6 Informatics luis rocha 2015 INDIANA UNIVERSITY biologically Inspired computing
lecture 6 -inspired Sections I485/H400 course outlook Assignments: 35% Students will complete 4/5 assignments based on algorithms presented in class Lab meets in I1 (West) 109 on Lab Wednesdays Lab 0 :
4452 Mathematical Modeling Lecture 13: Chaos and Fractals
Math Modeling Lecture 13: Chaos and Fractals Page 1 442 Mathematical Modeling Lecture 13: Chaos and Fractals Introduction In our tetbook, the discussion on chaos and fractals covers less than 2 pages.
vii Contents 7.5 Mathematica Commands in Text Format 7.6 Exercises
Preface 0. A Tutorial Introduction to Mathematica 0.1 A Quick Tour of Mathematica 0.2 Tutorial 1: The Basics (One Hour) 0.3 Tutorial 2: Plots and Differential Equations (One Hour) 0.4 Mathematica Programs
DYNAMICAL SYSTEMS
0.42 DYNAMICAL SYSTEMS Week Lecture Notes. What is a dynamical system? Probably the best way to begin this discussion is with arguably a most general and yet least helpful statement: Definition. A dynamical
Geometric Limits of Julia Sets of Maps z^n + exp(2πiθ) as n
Butler University Digital Commons @ Butler University Scholarship and Professional Work - LAS College of Liberal Arts & Sciences 2015 Geometric Limits of Julia Sets of Maps z^n + exp(2πiθ) as n Scott R.
AN INTRODUCTION TO FRACTALS AND COMPLEXITY
AN INTRODUCTION TO FRACTALS AND COMPLEXITY Carlos E. Puente Department of Land, Air and Water Resources University of California, Davis http://puente.lawr.ucdavis.edu 2 Outline Recalls the different kinds
Revista Economica 65:6 (2013)
INDICATIONS OF CHAOTIC BEHAVIOUR IN USD/EUR EXCHANGE RATE CIOBANU Dumitru 1, VASILESCU Maria 2 1 Faculty of Economics and Business Administration, University of Craiova, Craiova, Romania 2 Faculty of Economics
Repeat tentative ideas from earlier - expand to better understand the term fractal.
Fractals Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. (Mandelbrot, 1983) Repeat tentative ideas from
The Mandelbrot Set. Andrew Brown. April 14, 2008
The Mandelbrot Set Andrew Brown April 14, 2008 The Mandelbrot Set and other Fractals are Cool But What are They? To understand Fractals, we must first understand some things about iterated polynomials
By Nadha CHAOS THEORY
By Nadha CHAOS THEORY What is Chaos Theory? It is a field of study within applied mathematics It studies the behavior of dynamical systems that are highly sensitive to initial conditions It deals with
Inaccessibility and undecidability in computation, geometry, and dynamical systems
Physica D 155 (2001) 1 33 Inaccessibility and undecidability in computation, geometry, and dynamical systems Asaki Saito a,, Kunihiko Kaneko b a Laboratory for Mathematical Neuroscience, Brain Science
Fractal dimensions of selected coastal water bodies in Kerala, SW coast of India - A case study
Indian Journal of Marine Sciences Vol. 36(2), June 2007, pp. 162-166 Fractal dimensions of selected coastal water bodies in Kerala, SW coast of India - A case study *Srikumar Chattopadhyay & S. Suresh
LINEAR CHAOS? Nathan S. Feldman
LINEAR CHAOS? Nathan S. Feldman In this article we hope to convience the reader that the dynamics of linear operators can be fantastically complex and that linear dynamics exhibits the same beauty and
Fractals from Simple Polynomial Composite Functions. This paper describes a method of generating fractals by composing
Fractals from Simple Polynomial Composite Functions Ken Shirriff 571 Evans Hall University of California Berkeley, CA 94720 ABSTRACT This paper describes a method of generating fractals by composing two
Multifractal Models for Solar Wind Turbulence
Multifractal Models for Solar Wind Turbulence Wiesław M. Macek Faculty of Mathematics and Natural Sciences. College of Sciences, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland;
FRACTALS LINDENMAYER SYSTEMS. November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin
FRACTALS LINDENMAYER SYSTEMS November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin RECAP HIDDEN MARKOV MODELS What Letter Is Written Here? What Letter Is Written Here? What Letter Is Written Here? The Idea
AN INTRODUCTION TO FRACTALS AND COMPLEXITY
AN INTRODUCTION TO FRACTALS AND COMPLEXITY Carlos E. Puente Department of Land, Air and Water Resources University of California, Davis http://puente.lawr.ucdavis.edu 2 Outline Recalls the different kinds
A Very Brief and Shallow Introduction to: Complexity, Chaos, and Fractals. J. Kropp
A Very Brief and Shallow Introduction to: Complexity, Chaos, and Fractals J. Kropp Other Possible Titles: Chaos for Dummies Learn Chaos in 1 hour All you need to know about Chaos Definition of Complexity
Control and synchronization of Julia sets of the complex dissipative standard system
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 4, 465 476 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.4.3 Control and synchronization of Julia sets of the complex dissipative standard system
Een vlinder in de wiskunde: over chaos en structuur
Een vlinder in de wiskunde: over chaos en structuur Bernard J. Geurts Enschede, November 10, 2016 Tuin der Lusten (Garden of Earthly Delights) In all chaos there is a cosmos, in all disorder a secret
Snowflakes. Insight into the mathematical world. The coastline paradox. Koch s snowflake
Snowflakes Submitted by: Nadine Al-Deiri Maria Ani Lucia-Andreea Apostol Octavian Gurlui Andreea-Beatrice Manea Alexia-Theodora Popa Coordinated by Silviana Ionesei (teacher) Iulian Stoleriu (researcher)
Analysis of Dynamical Systems
2 YFX1520 Nonlinear phase portrait Mathematical Modelling and Nonlinear Dynamics Coursework Assignments 1 ϕ (t) 0-1 -2-6 -4-2 0 2 4 6 ϕ(t) Dmitri Kartofelev, PhD 2018 Variant 1 Part 1: Liénard type equation
Fractal Geometry and Programming
Çetin Kaya Koç http://koclab.cs.ucsb.edu/teaching/cs192 koc@cs.ucsb.edu Çetin Kaya Koç http://koclab.cs.ucsb.edu Fall 2016 1 / 35 Helge von Koch Computational Thinking Niels Fabian Helge von Koch Swedish
Dynamical Systems: Lecture 1 Naima Hammoud
Dynamical Systems: Lecture 1 Naima Hammoud Feb 21, 2017 What is dynamics? Dynamics is the study of systems that evolve in time What is dynamics? Dynamics is the study of systems that evolve in time a system
PREDICTION OF NONLINEAR SYSTEM BY CHAOTIC MODULAR
Journal of Marine Science and Technology, Vol. 8, No. 3, pp. 345-35 () 345 PREDICTION OF NONLINEAR SSTEM B CHAOTIC MODULAR Chin-Tsan Wang*, Ji-Cong Chen**, and Cheng-Kuang Shaw*** Key words: chaotic module,
Chaos and Liapunov exponents
PHYS347 INTRODUCTION TO NONLINEAR PHYSICS - 2/22 Chaos and Liapunov exponents Definition of chaos In the lectures we followed Strogatz and defined chaos as aperiodic long-term behaviour in a deterministic
Julia Sets Converging to the Filled Basilica
Robert Thÿs Kozma Professor: Robert L. Devaney Young Mathematicians Conference Columbus OH, August 29, 2010 Outline 1. Introduction Basic concepts of dynamical systems 2. Theoretical Background Perturbations
Fractals. R. J. Renka 11/14/2016. Department of Computer Science & Engineering University of North Texas. R. J. Renka Fractals
Fractals R. J. Renka Department of Computer Science & Engineering University of North Texas 11/14/2016 Introduction In graphics, fractals are used to produce natural scenes with irregular shapes, such
Chaos theory. Applications. From Wikipedia, the free encyclopedia
Chaos theory From Wikipedia, the free encyclopedia Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy.
Abstract. 1 Introduction
Time Series Analysis: Mandelbrot Theory at work in Economics M. F. Guiducci and M. I. Loflredo Dipartimento di Matematica, Universita di Siena, Siena, Italy Abstract The consequences of the Gaussian hypothesis,
The Research of Railway Coal Dispatched Volume Prediction Based on Chaos Theory
The Research of Railway Coal Dispatched Volume Prediction Based on Chaos Theory Hua-Wen Wu Fu-Zhang Wang Institute of Computing Technology, China Academy of Railway Sciences Beijing 00044, China, P.R.
Abstract: Complex responses observed in an experimental, nonlinear, moored structural
AN INDEPENDENT-FLOW-FIELD MODEL FOR A SDOF NONLINEAR STRUCTURAL SYSTEM, PART II: ANALYSIS OF COMPLEX RESPONSES Huan Lin e-mail: linh@engr.orst.edu Solomon C.S. Yim e-mail: solomon.yim@oregonstate.edu Ocean
Dynamical Systems with Applications
Stephen Lynch Dynamical Systems with Applications using MATLAB Birkhauser Boston Basel Berlin Preface xi 0 A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox 1 0.1 Tutorial One: The Basics
PHY100 One Last Time Homepage. Current Assignments... Writing Assignment #1. Writing and Plagiarism 24/01/2013
Fractal Broccoli Copyright: Jon Sullian, PDPhoto.org http://en.wikipedia.org/wiki/file:fractal_broccoli.jpg Chaos is a friend of mine. Bob Dylan, American folksinger (1941-) Predictability: Does the flap
Chaotic transport through the solar system
The Interplanetary Superhighway Chaotic transport through the solar system Richard Taylor rtaylor@tru.ca TRU Math Seminar, April 12, 2006 p. 1 The N -Body Problem N masses interact via mutual gravitational
Chaos in GDP. Abstract
Chaos in GDP R. Kříž Abstract This paper presents an analysis of GDP and finds chaos in GDP. I tried to find a nonlinear lower-dimensional discrete dynamic macroeconomic model that would characterize GDP.
Automorphic Equivalence Within Gapped Phases
1 Harvard University May 18, 2011 Automorphic Equivalence Within Gapped Phases Robert Sims University of Arizona based on joint work with Sven Bachmann, Spyridon Michalakis, and Bruno Nachtergaele 2 Outline:
Infinity Unit 2: Chaos! Dynamical Systems
Infinity Unit 2: Chaos! Dynamical Systems Iterating Linear Functions These questions are about iterating f(x) = mx + b. Seed: x 1. Orbit: x 1, x 2, x 3, For each question, give examples and a symbolic
LECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:
Lecture 6 Chaos Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Chaos, Attractors and strange attractors Transient chaos Lorenz Equations
Select / Special Topics in Classical Mechanics. P. C. Deshmukh
STiCM Select / Special Topics in Classical Mechanics P. C. Deshmukh Department of Physics School of Basic Sciences Indian Institute of Technology Madras Indian Institute of Technology Mandi Chennai 600036
Fractals: How long is a piece of string?
Parabola Volume 33, Issue 2 1997) Fractals: How long is a piece of string? Bruce Henry and Clio Cresswell And though the holes were rather small, they had to count them all. Now they know how many holes
Chaos Theory and Nonsqueezed-State Boson-Field Mechanics in Quantum Gravity and Statistical Optics. Kevin Daley
Chaos Theory and Nonsqueezed-State Boson-Field Mechanics in Quantum Gravity and Statistical Optics Kevin Daley April 25, 2009 Abstract: Previous work, resulting in an entropy-based quantum field theory
Dynamical Systems with Applications using Mathematica
Stephen Lynch Dynamical Systems with Applications using Mathematica Birkhäuser Boston Basel Berlin Contents Preface xi 0 A Tutorial Introduction to Mathematica 1 0.1 A Quick Tour of Mathematica 2 0.2 Tutorial
INTRODUCTION TO CHAOS THEORY T.R.RAMAMOHAN C-MMACS BANGALORE
INTRODUCTION TO CHAOS THEORY BY T.R.RAMAMOHAN C-MMACS BANGALORE -560037 SOME INTERESTING QUOTATIONS * PERHAPS THE NEXT GREAT ERA OF UNDERSTANDING WILL BE DETERMINING THE QUALITATIVE CONTENT OF EQUATIONS;
Julia Sets and the Mandelbrot Set
Julia Sets and the Mandelbrot Set Julia sets are certain fractal sets in the complex plane that arise from the dynamics of complex polynomials. These notes give a brief introduction to Julia sets and explore
Entropies and fractal dimensions
Entropies and fractal dimensions Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Entropies and fractal dimensions. Philica, Philica, 2016. HAL Id: hal-01377975
On accessibility of hyperbolic components of the tricorn
1 / 30 On accessibility of hyperbolic components of the tricorn Hiroyuki Inou (Joint work in progress with Sabyasachi Mukherjee) Department of Mathematics, Kyoto University Inperial College London Parameter
ONE DIMENSIONAL CHAOTIC DYNAMICAL SYSTEMS
Journal of Pure and Applied Mathematics: Advances and Applications Volume 0 Number 0 Pages 69-0 ONE DIMENSIONAL CHAOTIC DYNAMICAL SYSTEMS HENA RANI BISWAS Department of Mathematics University of Barisal
MTH4100 Calculus I. Week 8 (Thomas Calculus Sections 4.1 to 4.4) Rainer Klages. School of Mathematical Sciences Queen Mary, University of London
MTH4100 Calculus I Week 8 (Thomas Calculus Sections 4.1 to 4.4) Rainer Klages School of Mathematical Sciences Queen Mary, University of London Autumn 2008 R. Klages (QMUL) MTH4100 Calculus 1 Week 8 1 /
Complex Analysis for F2
Institutionen för Matematik KTH Stanislav Smirnov stas@math.kth.se Complex Analysis for F2 Projects September 2002 Suggested projects ask you to prove a few important and difficult theorems in complex
Fractal trajectories of the dynamical system
Fractal trajectories of the dynamical system Mare Berezowsi Silesian University of Technology Institute of Mathematics Gliwice, Poland E-mail address: mare.berezowsi@polsl.pl Abstract The scope of the
JULIA SETS AND BIFURCATION DIAGRAMS FOR EXPONENTIAL MAPS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, July 1984 JULIA SETS AND BIFURCATION DIAGRAMS FOR EXPONENTIAL MAPS BY ROBERT L. DEVANEY ABSTRACT. We describe some of the
Periodic Sinks and Observable Chaos
Periodic Sinks and Observable Chaos Systems of Study: Let M = S 1 R. T a,b,l : M M is a three-parameter family of maps defined by where θ S 1, r R. θ 1 = a+θ +Lsin2πθ +r r 1 = br +blsin2πθ Outline of Contents:
A Comparison of Fractal Dimension Algorithms by Hurst Exponent using Gold Price Time Series
ISS: 3-9653; IC Value: 45.98; SJ Impact Factor: 6.887 Volume 6 Issue II, February 08- Available at www.ijraset.com A Comparison of Fractal Dimension Algorithms by Hurst Exponent using Gold Price Time Series
B5.6 Nonlinear Systems
B5.6 Nonlinear Systems 1. Linear systems Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Linear systems 1.1 Differential Equations 1.2 Linear flows 1.3 Linear maps
Siegel disk for complexified Hénon map
Siegel disk for complexified Hénon map O.B. Isaeva, S.P. Kuznetsov arxiv:0804.4238v1 [nlin.cd] 26 Apr 2008 Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov, 410019,
Chapter 3. Gumowski-Mira Map. 3.1 Introduction
Chapter 3 Gumowski-Mira Map 3.1 Introduction Non linear recurrence relations model many real world systems and help in analysing their possible asymptotic behaviour as the parameters are varied [17]. Here
Transform Solutions to LTI Systems Part 3
Transform Solutions to LTI Systems Part 3 Example of second order system solution: Same example with increased damping: k=5 N/m, b=6 Ns/m, F=2 N, m=1 Kg Given x(0) = 0, x (0) = 0, find x(t). The revised
Mathematics Specialist Units 3 & 4 Program 2018
Mathematics Specialist Units 3 & 4 Program 018 Week Content Assessments Complex numbers Cartesian Forms Term 1 3.1.1 review real and imaginary parts Re(z) and Im(z) of a complex number z Week 1 3.1. review
The Sine Map. Jory Griffin. May 1, 2013
The Sine Map Jory Griffin May, 23 Introduction Unimodal maps on the unit interval are among the most studied dynamical systems. Perhaps the two most frequently mentioned are the logistic map and the tent
Fractals. Part 6 : Julia and Mandelbrot sets, Martin Samuelčík Department of Applied Informatics
Fractals Part 6 : Julia and Mandelbrot sets, Martin Samuelčík Department of Applied Informatics Problem of initial points Newton method for computing root of function numerically Computing using iterations
Fractals. Krzysztof Gdawiec.
Fractals Krzysztof Gdawiec kgdawiec@ux2.math.us.edu.pl 1 Introduction In the 1970 s Benoit Mandelbrot introduced to the world new field of mathematics. He named this field fractal geometry (fractus from
Double Transient Chaotic Behaviour of a Rolling Ball
Open Access Journal of Physics Volume 2, Issue 2, 2018, PP 11-16 Double Transient Chaotic Behaviour of a Rolling Ball Péter Nagy 1 and Péter Tasnádi 2 1 GAMF Faculty of Engineering and Computer Science,
Non Locally-Connected Julia Sets constructed by iterated tuning
Non Locally-Connected Julia Sets constructed by iterated tuning John Milnor Stony Brook University Revised May 26, 2006 Notations: Every quadratic map f c (z) = z 2 + c has two fixed points, α and β, where
Can two chaotic systems make an order?
Can two chaotic systems make an order? J. Almeida, D. Peralta-Salas *, and M. Romera 3 Departamento de Física Teórica I, Facultad de Ciencias Físicas, Universidad Complutense, 8040 Madrid (Spain). Departamento
I NONLINEAR EWORKBOOK
I NONLINEAR EWORKBOOK Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and SymbolicC++ Programs Willi-Hans Steeb
Complicated dynamics from simple functions
Complicated dynamics from simple functions Math Outside the Box, Oct 18 2016 Randall Pyke rpyke@sfu.ca This presentation: www.sfu.ca/~rpyke Presentations Dynamics Discrete dynamical systems: Variables
THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
Concepts in Theoretical Physics. John D Barrow. Lecture 1: Equations in Physics, Simplicity, and Chaos
Concepts in Theoretical Physics Lecture 1: Equations in Physics, Simplicity, and Chaos John D Barrow If people do not believe that mathematics is simple, it is only because they do not realize how complicated
Hands-on table-top science
Hands-on table-top science Harry L. Swinney University of Texas fluid convection bacterial colony chemical pattern daffodil 5 cm A special thanks to the students and postdocs at U. Texas who did this work
Nonsmooth systems: synchronization, sliding and other open problems
John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England Nonsmooth systems: synchronization, sliding and other open problems 2 Nonsmooth Systems 3 What is a nonsmooth
Effect of various periodic forces on Duffing oscillator
PRAMANA c Indian Academy of Sciences Vol. 67, No. 2 journal of August 2006 physics pp. 351 356 Effect of various periodic forces on Duffing oscillator V RAVICHANDRAN 1, V CHINNATHAMBI 1, and S RAJASEKAR
Simple approach to the creation of a strange nonchaotic attractor in any chaotic system
PHYSICAL REVIEW E VOLUME 59, NUMBER 5 MAY 1999 Simple approach to the creation of a strange nonchaotic attractor in any chaotic system J. W. Shuai 1, * and K. W. Wong 2, 1 Department of Biomedical Engineering,
Weak Mean Stability in Random Holomorphic Dynamical Systems Hiroki Sumi Graduate School of Human and Environmental Studies Kyoto University Japan
Weak Mean Stability in Random Holomorphic Dynamical Systems Hiroki Sumi Graduate School of Human and Environmental Studies Kyoto University Japan E-mail: sumi@math.h.kyoto-u.ac.jp http://www.math.h.kyoto-u.ac.jp/
Attractor of a Shallow Water Equations Model
Thai Journal of Mathematics Volume 5(2007) Number 2 : 299 307 www.math.science.cmu.ac.th/thaijournal Attractor of a Shallow Water Equations Model S. Sornsanam and D. Sukawat Abstract : In this research,
Wave propagation and attenuation in wool felt
Wave propagation and attenuation in wool felt Dmitri Kartofelev, Anatoli Stulov Laboratory of Nonlinear Dynamics, Institute of Cybernetics at Tallinn University of Technology, Tallinn, Estonia. Summary
FRACTAL GEOMETRY, DYNAMICAL SYSTEMS AND CHAOS
FRACTAL GEOMETRY, DYNAMICAL SYSTEMS AND CHAOS MIDTERM SOLUTIONS. Let f : R R be the map on the line generated by the function f(x) = x 3. Find all the fixed points of f and determine the type of their