Lecture 6 - Thursday, 24 January 2013

• Introduction to chaos
• The three-body gravitational problem
• The pendulum as an attractor
• Lorentz attractors
• Fractals

Powerpoint Slides

Note: These slides do not include all material presented in class - additional material was presented on the blackboard and in demonstrations.

PDF Format - 1 slide per page

PDF Format - 6 slides per page

Supplementary Material

Today's demonstrations: (i) Double pendulum; (ii) Simple pendulum; (iii) Bowling ball pendulum; (iv) Driven mass on a spring

Here's the book I recommended in class "Chaos: Making a New Science" by James Gleick: http://www.around.com/chaos.html

Today's animations: (I showed some of these in class. Try them all at home.)

• Flash animation of the gravitational three-body problem - try it out: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/Chaos/ThreeBody/ThreeBody.html

• Flash animation of two solutions to the gravitational three-body problem: http://www.upscale.utoronto.ca/PVB/Harrison/Chaos/anim3body2.html

• Illustration of the Butterfly Effect - try it out: http://www.exploratorium.edu/complexity/java/lorenz.html

• Java applet of pendulum motion - make your own stable or strange attractor: http://www.physics.orst.edu/~rubin/nacphy/JAVA_pend/COMP/

• 3D movie of the Lorenz Attractor: http://paulbourke.net/fractals/lorenz/lorenz.m4v

• More on the Lorenz Attractor - movie, images, computer codes: http://paulbourke.net/fractals/lorenz/

• Flash animation of the Lorentz attractor: http://www.upscale.utoronto.ca/PVB/Harrison/Chaos/animlorenz.html

• Interactive animation of the Lorentz attractor: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/Chaos/Lorenz/Lorenz.html

• Animation of Koch Snowflakes: http://www.absorblearning.com/media/attachment.action?quick=mf&att=1606

• Infinite magnification of Sierpinski Triangles: http://math.bu.edu/DYSYS/animations/sierp-zoom.html

• Interactive animation of the Mandelbrot Set: http://www.ph.biu.ac.il/~rapaport/java-apps/mandel.html

• Chaos and Fractals - a concise overview of some of the topics we are discussing: http://www.pha.jhu.edu/~ldb/seminar/index.html

• Wikipedia on chaos theory - a good overview with lots of references: http://en.wikipedia.org/wiki/Chaos_theory

• The Chaos Hypertextbook - for those who really want explore the math and physics of chaos theory: http://hypertextbook.com/chaos/

• And also from the Chaos Hypertextbook - loads of internet resources: http://hypertextbook.com/chaos/93.shtml

• The Open Directory on chaos and fractals - lots of links: http://www.dmoz.org/Science/Math/Chaos_and_Fractals/

• One of the many websites showing beautiful fractals: http://www.miqel.com/fractals_math_patterns/visual-math-iterative-fractals.html

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This web site is maintained by Kimberly Strong.
Last updated January 24, 2013.