Chaotic System logo Title: Chaotic System Demonstrators
(and some other complex systems stuff)
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Note that this page includes some very useful Flash apps (which may not run on many Apple systems).

Lorenz Attractor/"strange attractor"

A Lorenz 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.)

Lorenz Attractor Demo

And there's a more modern demonstrator at

Einstein's Explanation of Brownian Motion

animated demo 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 NOTE: To see the Java applet, you'll need to have the Java plug-in working on your web-browser.

Double Pendulum

Click the image above to go to an .mp4 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.

Double pendulum | Chaos | Butterfly effect | Computer simulation

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.

Double Pendulum for sale

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). (direct link).

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.)

Clausewitzian "Trinity" demonstration device

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 (USA).

Randomly Oscillating Magnetic Pendulum

Logistic Map/Bifurcation Diagram

Logistic Map

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 You'll have to install it on your own computer, of course.

See also

A Complex Adaptive System (CAS)

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

Eric Berlow: "How Complexity Leads to Simplicity"
TED Global, 2010. Filmed July 2010. Posted on TED November 2010.
Original Url

Click the image below for a larger, reasonably high-resolution version of the Afghanistan graphic to which speaker Eric Berlow refers.

insanely complicated flow-chart of security issues in Afghanstan

(on Complexity broadly speaking, including the narrower field of Chaos)

James Gleick discussing his famous book Chaos: Making a New Science (New York: Penguin, 1987). Get the 20th anniversary edition (New York: Penguin, 2008), ISBN 9780143113454.

Video image

DVDNOVA: "Hunting the Hidden Dimension."This is a 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. Also available on DVD from

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 There is a low-bandwidth version of the video here.


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. Great graphics and animations; the pace is oddly slow.

Film-Chapter1 CHAOS I: Motion and Determinism
—Panta Rhei

Film-Chapter2 CHAOS II: Vector Fields
—The Lego Race

Film-Chapter3 CHAOS III: Some Mechanics
—The Apple and the Moon

Film-Chapter4 CHAOS IV: Oscillations
—The Swing

Film-Chapter5 CHAOS V: Billiards
—Duhem's Bull

Film-Chapter6 CHAOS VI: Chaos and the Horseshoe
—Smale in Copacabana

Film-Chapter7 CHAOS VII: Strange Attractors
—The Butterfly Effect

Film-Chapter8 CHAOS VIII: Statistics
—Lorenz's Mill

Film-Chapter9 CHAOS IX: Chaotic or Not?
—Research Today


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