Chaos Theory

ThereIsNoSuchThingAsClimate?, maybe, which is a good argument for accepting organizational processes as patterns even though they are not "stable".
Interesting ThingsHappenAtBoundaries
Chaotic systems are SimilarAcrossScales?
General behavior of chaotic systems can be observed and documented, but exact behavior cannot be predicted
Systems can be categorized as explosive (expands unbounded), terminal (collapses to nothing), oscillating (mechanically cyclic and predictable), or stable (which isn't the right word, since such a system is internally chaotic).
Behaviors emerge in chaotic systems which cannot be predicted by examining the inputs to the system or transformations caused by the system.

[on the ButterflyEffect, copied here from TheTippingPoint page:]

The reason that the butterfly phenomena works is that the system in question is complex, with each element interacting with each other element in complex (nonlinear ways) so even slightly different initial conditions (this would be a snapshot of a particular moment) can result in very different results.

So one set of initial conditions has the butterfly with it's wings closed, no storm. Another, completely identical set of conditions, except that the butterflies wings are open results in a storm.

Because the only conditions that are different are the state of the butterflies wings, you might argue that the flap "caused" the storm.

The first difficulty with this explanation is that the system is so interdependent that it is impossible to prove anything effects anything else.

The second difficulty is that in order to prove that the wing flap caused the storm, you have to start out with absolutely the same initial conditions down to the infinite decimal place and do the calculation exactly (no rounding errors allowed).

Why? Because even slight changes can propagate to big differences. So any rounding error or difference in initial conditions can cause very different results, which would make it impossible to assign causation.

In the end, even if everything were done exactly there is a phenomena that guarantees you can't prove causation. Quantum fluctuation. In the initial conditions, there is a photon heading toward an atom. There is a probability that the photon will be absorbed, and one that it will not. As the system evolves, the effect of absorption versus non-absorption can spread. Enough of these events and the difference is felt.

In other words, even if the butterfly did not flap it's wings it still might storm, depending on the roll of a set of million-sided dice.

In other words the butterfly phenomenon is just a bad popularization. -- ThaddeusOlczyk

That just makes the state of the wings one cause among many others.

The idea that the flapping of a single butterfly's wings has large-scale consequences doesn't make much energetic sense. Such a small change can only have large-scale consequences if the global system is in an energetic minimum smaller than the energy of the butterfly's wings, that is to say, if the global system is in an energetically extremely unstable state, however, this can only be a very transitory moment. The chances that the butterfly's wings will flap at such a moment are vanishingly small. Maybe the butterfly-caused storm is just a BadAnalogy? set up to look spectacular in order to sell books. -- AndyPierce

This is incorrect and misunderstands the very definition of chaos theory. E.g. f you simulate the weather 5 months ahead, and then rerun the simulation with everything precisely the same except for the equivalent of a butterfly's wing gust of air, the results will typically be unrelated to the results of the first trial. (Which is approximately what Lorenz did.) The butterfly will, in general, have an arbitrarily large effect after a sufficiently long period of time passes. This is the defining characteristic of chaotic systems.
Neither statement is completely correct nor completely incorrect. Your rebuttal is partially correct, but Andy's statement is correct if qualified to apply to either (A) a change that happens in the basin of an attractor, which has super stable topology, or (B) if the point is to switch from a current orbit to another deliberately chosen orbit (distinguished by taking the opposite course on an upcoming known bifurcation), in which case it's true that the precise timing is critical.
The original characterization of chaos was in terms of the effects of minor randomness in initial conditions. More recently it has become importance to distinguish that from controlled chaos, where future behavior is in fact predictable, because orbital bifurcations are predicted and controlled at (before) each juncture. This method is in fact being used for extremely low energy orbital trajectories, see e.g. http://www.pupress.princeton.edu/titles/7687.html -- DougMerritt

Very different states can have equal energy content. For example, all previous states of the universe back to the Big Bang had the same energy content as the current universe. After all, energy is conserved.

Note that a chaotic system is in an extremely unstable state all the time, in fact, this is almost by definition true. The system will not remain long in this unstable state, but it will proceed to some other unstable state.

BrokenLink

-- StephanHouben

See ChaosToOrder

DaveOlson's book applying ChaosTheory and CatastropheTheory to SoftwareDevelopment: ExploitingChaos - ISBN 0442011121 .