Chaotic Systems Are Fundamentally Deterministic

Chaotic does not mean ultra-random as some people seem to think. Chaotic systems are fundamentally deterministic dynamical systems that have the property of exhibiting wildly different outcomes as a function of the initial conditions. Small errors in estimation/ measurement of initial conditions can result in unusably large differences in predicted results. If the prediction time scale is much smaller than the Lyapunov time, and initial conditions are measured with precision, predicted outcomes can have very good fidelity , and this is how they're tamed analytically.

"Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[19] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random."

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Reply to
Fred Bloggs
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See Strogatz for a not-mathematically-overbearing introduction; the basics of linear algebra and first-order ODEs on the calculus side should be good:

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Reply to
bitrex

The kicker is that chaotic systems are very powerful noise amplifiers, and there is a fundamental limit to the precision of the initial conditions.

If you have a frictionless pool table with perfectly elastic collisions, and break the rack normally, the position uncertainties of the balls grow exponentially with time. That is, a small error in aiming makes both balls go off at a slightly incorrect angle. That angular error grows linearly as the balls roll, causing the next collisions to be further off in aim. That causes more angular error, which turns into more aiming error on the next collisions, and so on.

I don't have it handy, but I recall reading a calculation that showed that after 30 seconds of this, the amplified Heisenberg uncertainty became larger than the diameter of a ball, meaning that you could no longer--even in principle--predict which balls would collide next.

Information is appearing in the universe at a very high rate all the time.

Like many chaotic systems, the motions of these ideal pool balls have constraints, such as that the total energy of the system is constant. The balls will bounce around and bounce around forever, but none of them can ever jump off the table and hit the ceiling. As in epidemics, exponential growth just applies to the early stages. ;)

Cheers

Phil Hobbs

Reply to
Phil Hobbs

I'm mainly concerned with short term (3 days) weather prediction. The European model streams real time weather data into the simulation continuously, which I assume means the initial conditions are constantly being reset. There use tens if not hundreds of thousands of real time sensor inputs, terrestrial and satellite. If a butterfly flaps its wings in the Himalayas, they know about it. The price they pay is 24 hour turnaround, but they are consistently dead on accurate. It's a gold standard of weather prediction right now.

Reply to
Fred Bloggs

That looks good, will take a look at it. All I know about the subject is from a non-linear control text from 15 years ago. Non-linear control dwells on chaotic systems quite a bit too.

Reply to
Fred Bloggs

Okay, but your thread title is completely misleading--if a system is deterministic, it doesn't stop after three days.

Cheers

Phil Hobbs

Reply to
Phil Hobbs

You do say some of the strangest things.

Reply to
Rick C

In real-life systems, social/economic/climactic, we have only crude knowledge of the element behavior, the couplings, or the initial state. People and clouds don't behave like planets.

People too used to power think they can understand, hence should control, societies and economics. They usually screw things up.

Reply to
jlarkin

Couldn't that happen? Like five balls converge on one and transfer all their momentum. That doesn't seem to violate any conservation rules.

Reply to
jlarkin

Sure. Look at which way the wind is blowing, check the temperature upwind from here, predict.

That doesn't scale.

Reply to
jlarkin

There's lots of real-life systems we DO have knowledge of. Nothing about 'social/economic/climactic' subsets of real-life is distinctive in this respect.

So, you aren't a monarchist? Inherited entitlements of political power is, indeed, not generally held in high regard nowadays.

Reply to
whit3rd

For momentum, each component is separately conserved. Five balls moving in the X-Y plane don't have Z-component momentum.

Reply to
whit3rd

Not if the cue ball didn't start out with enough energy to do that.

Other chaotic systems, such as globular star clusters, can eject parts at any speed.

Cheers

Phil Hobbs

Reply to
Phil Hobbs

'T'aint a momentum issue. The table can supply any Z-component required, and of course the initial momentum of the cue ball is enough to make it bounce--as any beginning pool player knows. ;)

Cheers

Phil Hobbs

Reply to
Phil Hobbs

If you use a cue ball with a different diameter than the other balls (not uncommon in quarter-to-play tables) then the table can supply momentum. Otherwise, it takes a fierce spin transfer at the cue or some kind of miracle to make that bounce.

Reply to
whit3rd

Soooooo? It has an infinite number of chances, and the compliance of the felt isn't necessarily zero.

Cheers

Phil Hobbs

Reply to
Phil Hobbs

Oh don't get technical. I hate it when people get technical.

Reply to
jlarkin

with things set 'optimally' you could sometimes see a third / fourth period doubling. Till the noise in the circuit (mostly from multipliers) washed out the signal. With a computer you can watch period doubling to whatever accuracy you like.

George H.

Reply to
George Herold

I was attracted by the subject, but I find it strange that this is considered something new.

Reply to
Tom Del Rosso
<snip>

Here, during the Wimbledon tennis tournament, the main court has a sliding roof which is closed when rain approaches, a procedure which takes a few minutes.

Every year, someone writes in to a paper complaining that the authorities have access to much better weather forecasts than are available to the general public (or the man on the Clapham Omnibus, the

156 in this case).
Reply to
Clive Arthur

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