some searching on google turned up a few sites, but all were written in esoteric mathematical lexicon and such, so…i turn to you fine gents.
Can someone explain what Chaos Theory in more ordinary terms?
Okay, a system is a group of interacting thingies. You can look at the state of a system at one time and then at another time.
With a non chaotic (linear) system, you change the initial state by a little bit, the final state will change a little bit, you change the intitial state a little bit more and the final state will change a little bit more.
With a chaotic system, you change the intitial state a little bit and you cant predict what the final state is like.
Weather is a chaotic system, if the temperature in Mozambique was 32C instead of 31C, you might get a cyclone in Tenessee 2 weeks later, if it were 33C, you might get hail etc.
You can’t predict very well based on the initial state what the final state will be.
Just to clarify, the term “theory” has slightly different meanings in math than in other fields of science. “Chaos theory” isn’t a specific theory that makes certain predictions. The term refers to the whole field of mathematics that deals with nonlinear systems, like Shalmanese explained.
If you’re really interested in the subject, I recomend James Gleik’s Chaos, a really good introduction for the layman.
Chaos theory encompasses a number of more or less related subjects like dynamic systems and fractal geometry.
To build a little on what Shalmanese wrote, chaotic is not the same as random. In a random system, you can’t predict anything at all. In a chaotic system, you can’t predict the state of the system at any given moment, but you can see general trends. Pour a little milk in a cup of tea. You should see a cloud-like formation emerge. There is no way for you to predict exactly what the cloud will look like, but you can however predict that it will look like a cloud. You could say that chaos theory is the study of such patterns.
I’d recommend going out and buying James Geleik’s book CHAOS.
Do you already know some examples of chaos in the everyday world? How about: ‘buzzing’ vibrations such as squealing bearings in your PC power supply fan, squalling brakes, rubbing your dry hands hard across an inflated balloon. Your heartbeat, where the time between beats is never identical. Population explosions of neighborhood insects. The structure of clouds. The weird sounds in the plumbing when the faucet is turned almost off. Wind gusts. And of course the crazy bouncing styrofoam bead on the loudspeaker cone when a pure tone is played.
Very brief description: It’s “chaos” when the math behind a real world phenomenon ends up describing a fractal.
For example, a Chaos motion might look very random or “noisy”, but in reality it has a frozen fractal pattern in the math describing it. The “randomness” appears because there are decision-points in the motion, and the outcome of those decisions is affected by extremely small inputs.
I’m not sure about that. While fractal systems and chaotic systems do have some similarities, they don’t necessarily go hand in hand at all.
Why is it that no chaos theory thread is ever missing the gushing math fnaboy?
Because chaos theory is a mathematical discipline. Would you prefer that the exposition be left to those outside the field?
I’m not able to write long posts today, but in a thread on Ian Malcolm’s philosophies in “Jurassic Park”, I gave a basic explanation. It’s in this forum, if you want to search for it.
I believe I heard somewhere (probably in a Chaos class at University 
that the equations governing chaotic systems are “extremely” non-linear, and therefore mostly insoluble using current techniques.
Is it the case that, if our mathematical techniques progressed to the point of being able to solve these non-linear equations analytically, we’d be able to predict the behavior of chaotic systems precisely?
DarrenS
Not necessarily. The famous weather model (the one that gave rise to the phrase “the butterfly effect”) was one in which the meteorologist STARTED OUT with equations–they were oversimplified in comparison with what you’d need to even begin to try to predict the weather, but they seemed to work well as a model. At a certain point, the guy decided to run a certain segment from a certain starting point again and continue from there. He looked up and typed in the values for all parameters as they had existed at that starting point and entered them and started the model again. And watched as the model diverged from what it had done the first time. The discrepancy? The decimal precision of the values as printed out was not fine enough to capture the exact values for the parameters.
The point of this being? Namely that even if we had all of the equations and the math to solve them, we’d have to have nearly infinitely precise measurements in order to know what we’d need to know in order to be able to predict anything. Something akin to knowing the temperature in degrees Celsius at the center of every single cubic foot’s worth of airspace and ground space, planetwide, up to the upper edges of the ionisphere, to six decimal points’ worth of accuracy. And air speed and direction. And pollen count and airborne particle count and average density and weight and reflectivity. And humidity. And so on. It just ain’t practical. If we had infinite resources and could actually create and deploy such measuring devices, we’d screw up what we’re measuring with the interfering presence of our measuring devices (they themselves would be a major factor in “creating” the weather at that point).
Yes and no. What’s always going to limit us is the fact that we can’t do infinite-precision computations, and that means that error will always be introduced into the system. You know what happens from there.
That’s the piece I was missing - our inability to measure the initial conditions with infinite precision. And of course a small error in the measurement of the initial conditions radically affects the state at time t1 >> t0. So that’s why my Differential Equations professor was so obsessed with the difference between linear and non-linear systems
It’s my understanding that “Chaotic system” means “system with a fractal attractor.” Do you know of a system which exhibits deterministic chaos, but where it’s attractor ISN’T a fractal?
Depends on what you mean by a fractal, bbeaty. Nothing in nature is truly a fractal, but rather fractal-like. You can invent a system that is as arbitrarily “un-fractal-like” as you want and still get a chaotic solution. However, I’m not sure whether the solution would be deterministic.
Something I don’t understand about chaos theory, and perhaps I am misinterpreting what is being written here, why is it accepted that a system is unpredictable and therefore random? Just because we are unable to predict what will happen in a chaotic system doesn’t necessarily mean that the outcome is random, just beyond our ability to measure. Does that make sense? Our lack of ability to take measurments fine enough to determine the outcome of an event should not preclude the possibility of a fixed outcome.
Certainly, there is a specific outcome that will fall within some specific parameters. But the point is that it is impossible to know what that outcome will be beforehand.
Chaos theory, I believe, is generally seen to have started when a meteorologist, Ed Lorenz, started studying a set of fairly simple equations in 1963 and found them to have very interesting characteristics.
One of them, and this has been mentioned before, is that tiny changes in the values of variables will have in the long run large and apparently unpredictable effects. This is not a question of measurements, it’s a fundamental characteristic of certain equations. The best way to understand Lorenz’ equations is to actually look at them.
You can’t look at the graph and predict exactly where the line’s going to go. However, let the aplet run long enough and you can clearly see a pattern emerging. Like I said in my post above chaotic != random.
Random is the opposite of Deterministic.
Chaotic is the opposite of Linear.
With a deterministic system, if you start off with EXACTLY the same start conditions, you will end up with the SAME finish conditions EVERY time. Of course, technically real world systems are either all completely determinsitic or completely random depending on which side of the quantum coin you fall on but we can talk about artifical systems as either being determinstic or random.
An example of a random system that in linear is dice throwing. Each dice throw is random but the chance of you getting a 6 is pretty much 1/6 throughout, no matter what your first throw was.
An eample of a deterministic system that is chaotic would be the three body problem. Stated simply, if you have 3 chunks off mass all orbiting around each other (eg. a small sun and two large planets) , tiny changes in the intitial positions/velocities will give you radically different positions/velocities if you let it run for a couple of thousand years. However, the system is completely dictated by gravity, theres no chance anywhere in the system.
Sorry, Shalmanese, but there is no randomness outside of quantum mechanics. If you could throw the dice in precisely the same way, you would get the same result each time. The problem is in the “if”.
The dice are not a linear system, either. The interactions with the ground are complex.
Your example of the three-body problem is correct: deterministic and chaotic.
That reminds me, I need to go check on my cat . . .