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




1.         Wolfram Research's web site World of Science, A Wolfram Web Resource at 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 “Physics,” then on “Mechanics,” 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  This page includes a simple animation.


1.2      The double pendulum is described rather mathematically at the URL  One can skip the mathematics and look at the graphs and the animation.


2.         The Wolfram Demonstration Project at contains instructive demonstrations. 


2.1.     Here is an opportunity to play with the fundamental concepts of simple harmonic motion:


2.2.     Here is a demonstration of the interesting behavior of a pendulum that is subjected to an external perturbing force:


2.3      This is an opportunity to play with the chaotic dynamics of a magnetic pendulum:


3.         The Archives of Mathematics has a home page at a page that links to many other resources.  Of particular use is the page Topics in Mathematics at  The topics “Dynamical Systems,” “Fractals,” and “Nonlinear Dynamics” list particularly useful links for our purposes.




4.         A rather good introduction with some illuminating animations is to be found at the web site 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  The illustrations are quite good at this site.





6.         David Harrison of the University of Toronto has created a site Flash Animations for Physics at - chaos.  This site contains instructive illustrations of concepts in classical mechanics and chaos.


6.1.     This may help one visualize simple harmonic motion:


6.2.     Here is a picture of simple harmonic motion:  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:


6.4.     Here is simple harmonic motion with friction:


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:


6.6      Here is a classic example of sensitivity to initial conditions: 


6.7      And this is an opportunity to play with (an abstract) problem in celestial mechanics.  The motion looks chaotic:




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


7.1      Here is a wonderful video on the most recent discoveries about the mechanism:                


7.1      With adequate computing power, one could look at the simulations and other images of the Antikythera Mechanism at




8.         A technical analysis of the double pendulum can be found at the site,  Near the bottom of this page, there is an animation of the chaotic motion of a double pendulum.


8.1      The web site 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




              Here are sites worth visiting.


9.         The Fractal Geometry of the Mandelbrot Set at and at


10.      Julia and Mandelbrot Set Explorer at




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


12.      Candidates for the impacting bodies are the Near Earth Objects (NEOs).  The NASA Nearth Earth Object Program is described at


13.      The Harvard-Smithsonian Center for Astrophysics operates the NEO Page at






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