Lorenz Attractor/"strange attractor"

Click on image

The "Butterfly Effect," or more technically, "sensitive dependence on initial conditions," is the essence of chaos. Clicking on the image above will take you to a Java applet at CalTech that will let you change the parameters of the system.

Another view:

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Einstein's Explanation of Brownian Motion

This animation demonstrates Brownian motion. Brownian motion is an old concept and does not really have anything to do with Chaos or Complexity per se—it really is random movement, though it can often look quite like it has some purpose or direction. The big particle can be considered as a dust particle while the smaller particles can be considered as molecules of a gas. On the left is the view one would see through a microscope. To the right is the physical explanation for the "random walk" of the dust particle. This is an animated gif made for computers unable to run Java. The original applet, credits, and a lecture on Brownian Motion can be found at http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html. NOTE: To see the Java applet, you'll need to have the Java plug-in working on your web-browser.

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The Three-Body Problem in Newtonian Physics

This FLASH animation is from "Flash Animations for Physics," by David M. Harrison, Department of Physics, University of Toronto. This is not a pre-drawn cartoon: Newton's equations are actually embedded in the Flash, which simply takes your settings and plots the mathematical output in graphic form. If you check the button for addional independent planets, you should nderstand that these are not really 'additional' planets—they are identical and don't exert any gravitational influence on one another. So what they really represent is the same planet in four slightly different starting positions. The differing trajectories they follow illustrate, in essence, the same mesage as the "Butterfly Effect": very small differences in input can have very large effects on output.

Different settings may reveal this effect in different ways and at different speeds. The default setting is pretty cool. Another setting that makes the larger point reasonably quickly:
Sun 1: 1.38
Sun 2: x initial = 0.4
Check button for 4 independent Planets

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Double Pendulum

Click the image above to go to a Flash animation of a double pendulum, an extremely simple system (there are only two points of freedom) that demonstrates deterministic chaos. This is an excellent way to demonstrate how complex behavior—even seemingly random behavior—can be created by very simple rules and systems.

Click the image above to go to a video of a live demonstration of a dual double pendulum. The double pendulum itself is an extremely simple system (there are only two points of freedom) that demonstrates deterministic chaos. The dual version (in which two double pendulums are suspended on the same axle) hows how dramatically the system's behavior can differ based on tiny differences in input (i.e., its sensitivity to initial conditions).

Here is a link to an interactive Java applet on the double pendulum from the Virtual Physics Laboratory at Northwestern University.

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Randomly Oscillating Magnetic Pendulum
(The image used in the Clausewitzian Trinity discussed in On War.)

Click on image

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Logistic Map/Bifurcation Diagram

This FLASH animation is from "Flash Animations for Physics," by David M. Harrison, Department of Physics, University of Toronto.

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The Game of Life

Invented in 1970 by John Horton Conway, Professor of Finite Mathematics at Princeton University. [This is discussed in Waldrop, Complexity--check the index for references. See also Daniel C. Dennett, Darwin's Dangerous Idea: Evolution and The Meanings of Life (New York: Simon & Schuster; Reprint edition, 1996), pp.166-176.] This simple mathematical exercise led investigators to a great many other ideas, but it also led directly to the entire field of "Artificial Life."

The Game of Life provides examples of "emergent complexity" and "self-organizing systems." It has a simple rule-set: "For each cell in the grid, count how many of its eight neighbors are ON at the present instant. If the answer is exactly two, the cell stays in its present state (ON or OFF) in the next instant. If the answer is precisely three, the cell is ON in the next instant whatever its current state. Under all other conditions, the cell is OFF." These basic rules were surprisingly hard to discover and seem to be unalterable or inevitable--that is, any other rules produce very uninteresting results: the system rapidly either freezes up solid or evaporates into nothingness. (Evidently, back in 1970 this game was more addictive to scientists than "Pong" was.)

Given this very simple rule (the only rule in the model universe we have created), the game demonstrates a surprising amount of complexity, depending on initial conditions. Click on these two screenshots from a computer-driven version of the game to see the very different system behaviors driven purely by the initial configuration of ON cells—um... What do we call that effect? (These are Flash videos. If Flash isn't working on your computer, try the .avi links.)

(.avi)   (.avi)

Play the game yourself. Various Game of Life applications can be downloaded free from http://www.ibiblio.org/lifepatterns/. You'll have to install it on your own computer, of course.

See also http://www.math.com/students/wonders/life/life.html.

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A Complex Adaptive System (CAS)

Although it may look like a street in India, this is actually the Westphalian state system.

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Another complex adaptive system at work?

Originally downloaded from: http://www.nmm.ac.uk/searchbin/searchs.pl?flashy=et1740z&flash=true.
See also: http://www.bbc.co.uk/history/interactive/animations/trafalgar/index_embed.shtml