**MLAP 31500**

**Natural Sciences Elective**

**Order and Chaos in the Natural World**

Spring Quarter 2014

Links to Useful and Interesting Web Sites

February 16, 2014

All of the links listed below appear to reach useful sources of information regarding dynamical systems and order and chaos in the natural world. There are many more web sites than can be listed here. Therefore, users are urged to exploit the links at the sites listed here and to search the web independently for other interesting sites. Any student who finds a particularly good site not listed here is invited to send me the URL via e-mail at voort@oddjob.uchicago.edu.

TWO GENERAL SITES

1. Wolfram Research's web site World of Science, A Wolfram Web Resource at http://scienceworld.wolfram.com/ covers many fields of mathematics, physics, and chemistry. Entries range from definitions of only a few lines to extended (often mathematical!) essays. In order to reach many entries relevant to order and chaos, click on ÒPhysics,Ó then on ÒMechanics,Ó and finally on a suitable key word. Biographical information on workers in many fields of science is also found at this site.

1.1 For information on the simple pendulum, including a phase portrait, see the URL http://scienceworld.wolfram.com/physics/Pendulum.html. This page includes a simple animation.

1.2 The double pendulum is described rather mathematically at the URL http://scienceworld.wolfram.com/physics/DoublePendulum.html. One can skip the mathematics and look at the graphs and the animation.

2. The Wolfram Demonstration Project at http://demonstrations.wolfram.com/ contains instructive demonstrations.

2.1. Here is an opportunity to play with the fundamental concepts of simple harmonic motion: http://demonstrations.wolfram.com/SimpleHarmonicMotionForASpring/.

2.2. Here is a demonstration of the interesting behavior of a pendulum that is subjected to an external perturbing force: http://demonstrations.wolfram.com/VibratingInvertedPendulum/.

2.3 This is an opportunity to play with the chaotic dynamics of a magnetic pendulum: http://demonstrations.wolfram.com/ChaoticDynamicsOfAMagneticPendulum/.

3. The
Archives of Mathematics has a home page at http://archives.math.utk.edu/ a page
that links to many other resources.
Of particular use is the page Topics in Mathematics at http://archives.math.utk.edu/topics/. The topics ÒDynamical Systems,Ó
ÒFractals,Ó and ÒNonlinear DynamicsÓ list particularly useful links for our
purposes.

INTRODUCTIONS TO CHAOS

4. A
rather good introduction with some illuminating animations is to be found at
the web site http://johnbanks.maths.latrobe.edu.au/chaos/ which presents Chaos: A pictorial introduction by a
mathematics group at La Trobe University.

5. An
even more comprehensive but still brief introduction to the subject is
contained in THE CHAOS HYPERTEXT BOOK by Glenn Elert at http://hypertextbook.com/chaos/. The illustrations are quite good at
this site.

PHYSICS
ANIMATIONS

6. David
Harrison of the University of Toronto has created a site Flash Animations for
Physics at http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/
- chaos. This site contains
instructive illustrations of concepts in classical mechanics and chaos.

6.1. This
may help one visualize simple harmonic motion: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/HookesLaw/HookesLaw.html.

6.2. Here
is a picture of simple harmonic motion: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/Circular2SHM/Circular2SHM.html. It illustrates the relation between
uniform circular motion in two dimensions and simple harmonic motion in one
dimension.

6.3. Simple
harmonic motion is ubiquitous: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/SHM/TwoSHM.html.

6.4. Here is simple harmonic motion with friction: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/DampedSHM/DampedSHM.html.

6.5. What happens when we apply a periodic driving force to a
damped simple harmonic oscillator?
Here is a toy with which we can play in order to fine out: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/DrivenSHM/DrivenSHM.html.

6.6 Here
is a classic example of sensitivity to initial conditions: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/Chaos/Bunimovich/Bunimovich.html.

6.7 And
this is an opportunity to play with (an abstract) problem in celestial
mechanics. The motion looks
chaotic: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/Chaos/ThreeBody/ThreeBody.html

THE ANTIKYTHERA MECHANISM

7. The
Antikythera Mechanism has been the subject of extensive
research in recent years. The Antikythera Mechanism Research Project offers a wealth of
information at http://www.antikythera-mechanism.gr/.

7.1 Here
is a wonderful video on the most recent discoveries about the mechanism: http://www.nature.com/nature/videoarchive/antikythera/.

7.1 With
adequate computing power, one could look at the simulations and other images of
the Antikythera Mechanism at http://etl.uom.gr/mr/Antikythera/index.html.

THE DOUBLE PENDULUM

8. A
technical analysis of the double pendulum can be found at the site http://scienceworld.wolfram.com/physics/DoublePendulum.html, Near the
bottom of this page, there is an animation of the chaotic motion of a double
pendulum.

8.1 The
web site http://www.clausewitz.com/Flash/FLVs/DualDblPendulumVid.htm
contains an instructive demonstration of a double pendulum.

8.2 M.
Kawaka has put a two simulations of a double pendulum
at the web page at http://www6.ocn.ne.jp/~simuphys/niju-furikoE.html.

CHAOS AND FRACTALS

Here
are sites worth visiting.

9. The
Fractal Geometry of the Mandelbrot Set at http://math.bu.edu/DYSYS/FRACGEOM/index.html
and at http://math.bu.edu/DYSYS/FRACGEOM2/FRACGEOM2.html.

10. Julia
and Mandelbrot Set Explorer at http://aleph0.clarku.edu/~djoyce/julia/explorer.html.

SMALL
BODIES IN THE SOLAR SYSTEM AND TERRESTRIAL IMPACTS

11. There
is a substantial geological record of impacts of small bodies (asteroids and/or
comets) with Earth. The University
of New Brunswick runs a database with relevant links at http://www.unb.ca/passc/ImpactDatabase/.

12. Candidates
for the impacting bodies are the Near Earth Objects (NEOs). The NASA Nearth
Earth Object Program is described at http://neo.jpl.nasa.gov/.

13. The Harvard-Smithsonian Center for Astrophysics operates the NEO Page at http://cfa-www.harvard.edu/iau/NEO/TheNEOPage.html.

**LINKS:**

Return to Course Page: mla315spring2013.html

Return to Peter Vandervoort's Home Page: pov.html

Go to the home page of the Department of Astronomy and Astrophysics

of the
University of Chicago: __ http://astro.uchicago.edu/__