Lorenz Attractor/"strange attractor"
Here (above) is as an animated .gif of Lorenz's "strange attractor." It describes a system very similar to Clausewitz's Trinity imagery, which has three attractors, but I find the Lorenz system to be especially relevant to Clausewitz's way of describing the variations in political and military ojjectives. (See the very informative Wikipedia article, "Lorenz System," from which the image above was borrrowed.)
And there's a more modern demonstrator at https://highfellow.github.io/lorenz-attractor/attractor.html.
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.
The Three-Body Problem in Newtonian Physics
*If you can't see a Flash application here (Thank you, Apple jerks), try reloading the page. It's pretty old tech and often clunky: try clicking around inside the area where it should be, or clicking around on any pieces of the image that may appear. Or try the original site. The larger image should look like this:
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, once again, 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
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. Focus on the behavior of the red dot. This is an excellent way to demonstrate how complex behavior—even seemingly random behavior—can be created by very simple rules and systems. (Click this link for a Flash version.)
This computer simulation shows how data generated by the pendulum can appear like totally random movement once you hide the mechanism itself. This illustrates the following quotation: "Don't tell me 'There's no right explanation.' There is a right explanation. We just don't know what it is." (Christopher Bassford)
Here's a very nice (and heavy) working double pendulum from Executive Model Design.
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's a Flash version)
Here is a link to an interactive Java applet on the double pendulum from the Virtual Physics Laboratory at Northwestern University.
Randomly Oscillating Magnetic Pendulum
(The image used in the Clausewitzian Trinity discussed in On War.)
The "Trinity" is a key concept in Clausewitzian theory, which Clausewitz illustrated by referring to this scientific device. You can obtain the ROMP (Randomly Oscillating Magnetic Pendulum) from science toy stores for about $30. This model is available from Amazon.com (USA).
Logistic Map/Bifurcation Diagram
*If you can't see a Flash application here, Thank you, Apple jerks. It's pretty old tech and often clunky, and it doesn't seem to like showing multiple Flashes on the same page. Try reloading the page, or clicking around inside the area where it should be, or clicking around on any pieces of the image that may appear. If that doesn't work, go to "Flash Animations for Physics," by David M. Harrison, Department of Physics, University of Toronto. The larger image should look like this:
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?)
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.
A Complex Adaptive System (CAS)
Although it may look like a street in India, this is actually the Westphalian state system.
Another complex adaptive system at work?
*If you can't see a Flash application here, Thank you, Apple jerks. It's pretty old tech and often clunky: try reloading the page, or clicking around inside the area where it should be, or clicking around on any pieces of the image that may appear. If that doesn't work, click this link. That one seems to work OK.]
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
Click the image below for a larger, reasonably high-resolution version of the Afghanistan graphic to which speaker Eric Berlow refers.
NOVA: "Hunting the Hidden Dimension." One-hour video on fractals, divided into five chapters. This documentary highlights a host of filmmakers, fashion designers, physicians, and other researchers who are using fractal geometry to innovate and inspire. Also lots of textual material. Original PBS Broadcast Date: October 28, 2008.
The Secret Life of Chaos (BBC 2010). Full Length Documentary. This video appears never to have been published by the BBC on DVD, but can be seen via the link above. You can find a great deal of video and information on it at http://atheistmovies.blogspot.com/2010/01/bbc-secret-life-of-chaos.html. There is a low-bandwidth version of the video here.
CHAOS: A MATHEMATICAL ADVENTURE
This is a film about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience. From Jos Leys, Étienne Ghys and Aurélien Alvarez, comes CHAOS, a math movie with nine 13-minute chapters. Available in a large choice of languages and subtitles. https://www.chaos-math.org/en/film.html. Great graphics and animations; the pace is oddly slow.