Chaos Theory (Part 2)

An attractor is a set in phase space that a particular dynamical system (weather patterns in the case of the Lorenz attractor) evolves towards over time. Any point which approaches the attractor will remain close to that attractor. The attractor can be a point, a line, a surface, or for many strange attractors, a fractal. That any strange attractor has a fractal set is to say that there exists some manifold with which the attractor can intersect to produce the Cantor set. A fractal is a geometric shape that exhibits both irregularity and self-similarity. Exhibiting self-similarity means that there are portions of the set, which if magnified a certain amount, is exactly the same set as the one which you magnified. Size-invariance is a continuous version of self-similarity. It must also exhibit irregularity, however, as a straight line is simple enough to be described by Euclidean geometry (e.g. high school geometry), and is not a fractal. Chaos theory typically deals with strange attractors. Pictures are really a much better way of understanding fractals and self similarity, so here are some cool ones!

First, the Cantor Set:












Other Famous Examples of Self-Similarity/Fractals

Mandelbrot Set














Mandelbrot Set's Correspondence with Logistic Map



















Sierpinski Triangle



















This site has a cool visualization:
http://serendip.brynmawr.edu/playground/sierpinski.html

Sierpinski Carpet



















Menger Sponge















& Mengerubik Cubesponge
http://forgetomori.com/2009/science/mengerubik-cubesponge/

Dragon Curve











Peano Curve
















Julia Set (of sin(z)...there are Julia sets of many functions)




















Koch Snowflake



















Koch Curve Self-Similarity








Brownian Tree (crystal growth)













Brownian Motion



















Lichtenberg Figures and Lightning




































(notice how it seems to choose one pathway after branching out - very interesting)


Ferns (real and fractal generated)





































Trees Too?
http://www.gskinner.com/blog/assets/InteractiveElm.html
http://www.webcalc.net/calc/0467.php

Artistic Representation of a Brain as Fractal



















Benoit Mandelbrot: Fractals and the art of roughness


Ron Eglash on African Fractals (from TED.com)


Another vid (visit the link):
PBS NOVA - Hunting the Hidden Dimension

- Watch more Videos at Vodpod.

Types of Strange Attractors



Lorenz Attractor














http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/Lorenz/
http://local.wasp.uwa.edu.au/~pbourke/fractals/lorenz/lorenz.m4v
http://crossgroup.caltech.edu/chaos_new/Lorenz.html

Julia Attractor (really cool!)
http://www.openprocessing.org/visuals/?visualID=8058

Rossler Attractor














Tamari Attractor



















Henon Attractor













Also look up: flow, Poincare mapping, bifurcations, catastrophe theory, self-organization

Fluid dynamics has also found much of chaos theory to be of use, especially in studies of turbulence, which I still consider to be unsolved.

A Karmen Vortex Street













Learn More
http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/
http://www.hypertextbook.com/chaos/
http://www.stsci.edu/~lbradley/seminar/index.html
http://www.scholarpedia.org/article/Chaos
http://www.calresco.org/ - a link of links
http://math.rice.edu/~lanius/frac/
http://www.imho.com/grae/chaos/chaos.html
http://www.societyforchaostheory.org/
http://www.exploratorium.edu/complexity/CompLexicon.html
http://www.csuohio.edu/sciences/dept/physics_from_www/kaufman/yurkon/chaos.html
http://classes.yale.edu/fractals/
http://www.egregium.us/
http://www.theory.org/

About Fractals and Fractal Art Specifically:
http://www.chaospro.de/
http://local.wasp.uwa.edu.au/~pbourke/fractals/
http://content.techrepublic.com.com/2346-10877_11-56937-1.html?tag=content;leftCol
http://www.enchgallery.com/
http://mitpress.mit.edu/books/FLAOH/cbnhtml/slides.html

More Intricate Articles and Research:
http://physics.mercer.edu/petepag/combow.html
http://www.zfm.ethz.ch/~leine/toys.htm
http://vigo.ime.unicamp.br/2p/PendulaProject.html
http://www.math.bme.hu/~bnc/wada/
http://www.matternews.com/research/Researchers_move_closer_to_understanding_chaotic_motion_of_a_solid_body_in_a_fluid.asp
http://www.osti.gov/accomplishments/prigogine.html

College Web Pages
http://www.santafe.edu/
http://www.cscs.umich.edu/
http://wwwrsphysse.anu.edu.au/nonlinear/
http://www-chaos.umd.edu/

Complexity in Social Science
http://www.irit.fr/COSI/

Catastrophe Theory
http://en.wikipedia.org/wiki/Catastrophe_theory
http://l.d.v.dujardin.pagesperso-orange.fr/ct/eng_index.html

Most Links I have found on this subject are incorrect, misleading, incorrect interpretations, disconnected, or spiritual. I utilized Lorenz's 'The Essence of Chaos', Gleick's 'Chaos: Making a New Science', and Schroeder's 'Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise' as references for my own writings. I didn't intend to copy anything directly, but I apologize if I did. Some of it may be due to the fact that sometimes things can't be said precisely without copying. I don't claim to have come up with any of the actual ideas expressed in either part of this post. There was just so much cool stuff out there on this subject I could hardly get myself to finish this article! Please check out the links and explore this more. I think chaos theory will yield very interesting physics research in years to come. I took many videos and pictures from all over the web. If I am violating any copyright laws or anything of the sort, please let me know and I will remove whatever is necessary. Also, if anybody finds incorrect statements with particular things which have been said, please let me know as I have tried very hard to express these concepts accurately.