Fractional Dimensions, Strange Attractors & Chaos

Slides:



Advertisements
Similar presentations
The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually.
Advertisements

Fractal Euclidean RockCrystal Single planet Large-scale distribution of galaxies.
Pendulum without friction
Fractals Part 1: In which we attempt to describe what a fractal is, because people will ask (but perhaps they will not really want to know the answer).
Chaos, Communication and Consciousness Module PH19510 Lecture 15 Fractals.
Jochen Triesch, UC San Diego, 1 Iterative Maps and Chaos Goals: what is deterministic chaos? how does it relate to randomness?
Jochen Triesch, UC San Diego, 1 Rendering of the Mandelbrot set: perhaps the most famous fractal Fractals.
CHAOS THEORY and the butterfly effect English presentation By: Assaad Moawad.
Chapter 9: Recursive Methods and Fractals E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley Mohan Sridharan Based on Slides.
Lab 8: The onset of chaos Unpredictable dynamics of deterministic systems, often found in non-linear systems. Background of chaos Logistic Equation Non-linear.
1. 2 Class #26 Nonlinear Systems and Chaos Most important concepts  Sensitive Dependence on Initial conditions  Attractors Other concepts  State-space.
Chaos and System dynamics Leon Chang. Edward Lorenz In the early 1960's using a simple system of equations to model convection in the atmosphere, Edward.
Fractals Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts Director, Arts Technology Center University of New.
The infinitely complex… Fractals Jennifer Chubb Dean’s Seminar November 14, 2006 Sides available at
1 Class #27 Notes :60 Homework Is due before you leave Problem has been upgraded to extra-credit. Probs and are CORE PROBLEMS. Make sure.
Deterministic Chaos PHYS 306/638 University of Delaware ca oz.
Fractals Complex Adaptive Systems Professor Melanie Moses March
CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel.
Fractals.
Admin stuff. Questionnaire Name Math courses taken so far General academic trend (major) General interests What about Chaos interests you the most?
Applied Mathematics Complex Systems Fractals Fractal by Zhixuan Li.
A PowerPoint presentation brought to you by Christian Malone and Alissa Ousley.
AJITESH VERMA 1.  Dictionary meaning of chaos- state of confusion lack of any order or control.  Chaos theory is a branch of mathematics which studies.
1 GEM2505M Frederick H. Willeboordse Taming Chaos.
Structured Chaos: Using Mata and Stata to Draw Fractals
Chaos Theory and Fractals By Tim Raine and Kiara Vincent.
The Chaos Game.
Chaos Theory and Encryption
Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems Fichter,
Chaos and Strange Attractors
Fractional Dimension! Presentedby Sonali Saha Sarojini Naidu College for Women 30 Jessore Road, Kolkata
Details for Today: DATE:9 th December 2004 BY:Mark Cresswell FOLLOWED BY:Nothing Chaos 69EG3137 – Impacts & Models of Climate Change.
Fractal Dynamics in Physiology Alterations with Disease and Aging Presentation by Furkan KIRAÇ.
C T H H A E O O S R Y PatternsOfLife Chaos What is Chaos? Chaos, from a static view, is “pieces waiting to come together,” an inchoate pattern about.
Strange Attractors From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the Society for chaos theory in.
Fractals. In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately)
The Science of Complexity J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the First National Conference on Complexity.
Chaos Theory MS Electrical Engineering Department of Engineering
Mathematics Numbers: Percentages Science and Mathematics Education Research Group Supported by UBC Teaching and Learning Enhancement Fund Department.
Jochen Triesch, UC San Diego, 1 Motivation: natural processes unfold over time: swinging of a pendulum decay of radioactive.
David Chan TCM and what can you do with it in class?
Some figures adapted from a 2004 Lecture by Larry Liebovitch, Ph.D. Chaos BIOL/CMSC 361: Emergence 1/29/08.
Fractal Project Mariellen Hemmerling. Fractals “A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at.
Imagine you were playing around with Apophysis when some other GHP Math student student came up behind you and said “Gee that’s pretty! What is that a.
Introduction to Chaos by: Saeed Heidary 29 Feb 2013.
Governor’s School for the Sciences Mathematics Day 4.
Section 6.1 Images Viewing a Gallery of Fractals. Look for patterns.
Run Lola Run! Theme Chaos Theory/ Butterfly Effect
Dynamical Systems 4 Deterministic chaos, fractals Ing. Jaroslav Jíra, CSc.
CSE 423 Computer Graphics | MUSHFIQUR ROUF CSE423:
Fractals! Fractals are these crazy objects which stretch our understanding of shape and space, moving into the weird world of infinity. We will look at.
Fractals Ed Angel Professor Emeritus of Computer Science
 Introduction  Definition of a fractal  Special fractals: * The Mandelbrot set * The Koch snowflake * Sierpiński triangle  Fractals in nature  Conclusion.
Controlling Chaos Journal presentation by Vaibhav Madhok.
Management in complexity The exploration of a new paradigm Complexity theory and the Quantum Interpretation Walter Baets, PhD, HDR Associate Dean for Innovation.
Complexity Leadership Dynamical Systems & Leadership Jim Hazy July 19, 2007.
Fractals Cassi Blum.
Fractals.
A Primer on Chaos and Fractals Bruce Kessler Western Kentucky University as a prelude to Arcadia at Lipscomb University.
1 What did we learn before?. 2 line and segment generation.
Spencer Hart Advisor: Gus Hart
Chaos Analysis.
Fractals Project Natalie Rowe.
HONR 300/CMSC 491 Fractals (Flake, Ch. 5)
Handout #21 Nonlinear Systems and Chaos Most important concepts
HONR 300/CMSC 491 Fractals (Flake, Ch. 5)
Modeling of Biological Systems
HONR 300/CMSC 491 Fractals (Flake, Ch. 5)
By: Bahareh Taghizadeh
Localizing the Chaotic Strange Attractors of Multiparameter Nonlinear Dynamical Systems using Competitive Modes A Literary Analysis.
Presentation transcript:

Fractional Dimensions, Strange Attractors & Chaos

Old Familiar Faces Dimensions of some Familiar Figures

‘Weird’ Objects What about these objects?

How to ‘Measure’ dimensions? One gets N copies if one scales by a factor r The dimension ‘d’ is given by OR

The ‘Cantor Set’ This is the 1/3 Cantor Set Note here N=2 & r=3 Hence i.e. Cantor Set is 0.63 dimensional !!

The Koch Snowflake Note here N=4 & r=3 Hence i.e.

The Sierpinski Gasket Here N=3, r=2 Using We have

Fractals in Nature

Computer Generated Fractals I The ‘Julia Set’

Computer Generated Fractals II The ‘Mandelbrot Set’

The Butterfly Effect Flap of a butterfly’s wing in Rio de Janeiro causes a hurricane in Lahore Mathematically sensitivity of a system on initial conditions Think Billiards

The Logistic Map Very simple system exhibiting ‘chaos’ Can be a model for bacterial population ‘r’ can be thought of as net growth rate As ‘r’ varies one sees a drastic changes in behavior

As were increase r ……..

…. and …finally ……CHAOS Note sensitivity on IC System does NOT ‘settle down’ Unpredictable!! Where are the fractals?

The ‘Parameter Picture’ Choose different IC Run the system for long times Plot long time behavior for different ‘r’ The resulting picture has fractal structure!!

Lorenz System (Butterfly Effect) A simplified Weather Model For certain values of parameters is chaotic Q: Is our weather unpredictable?

What should you take away? Fractals are all around us There is an intrinsic link between chaotic systems and fractals Fractals can be generated easily on a computer Butterfly Effect was a cool movie!

Credits: Thank you wikipedia contributors for many of the figures Questions?? Credits: Thank you wikipedia contributors for many of the figures