And you can build a chaotic oscillator, with a few opamps, whose state is thoroughly scrambled in a hundred cycles. The planetary system is weakly nonlinear. Weather is strongly nonlinear.
What causes the transitions? Why doesn't an ice age, or a heat wave, latch?
Given the current state and the rules, chaotic systems are predictable in the short term. In the long term, all you can say is what the statistics might be. The *state* of a chaotic system is more and more uncertain as you project time points further out in the future. In a decently chaotic system, it's entirely unknowable after some time. Even floating point errors will totally scramble the state of a strongly nonlinear chaotic system simulation, even if you perfectly know the initial state and the dynamics.
My TDR deconvolution algorithm turns out to be a chaotic system. It's loads of fun to play with. We might even use it some day soon.
Not for the statistics of the system state. In the case of the Logistic Map, even after a billion iterations you can predict with certainty that its value will be between 0 and 1. The constraint is built in the equation.
I don't claim LM is a model of anything, definitely not a
*weather* model and even less I am intepreting it's value as the 'glaciation percentage'. It is just one chaotic system, and still its value does not suddenly shoot to one million. Heck, it does not even shoot to 1.5 ! The LM has a level of predictability in it.
Let's look at the temperature in Chicago, which you mentioned - say, in July. It varies in an inpredictable manner, but remains roughly between +7.2 C and +40.6 C . The constraint is not as hard as with the LM, but experimentally the constraint seems to be there. It's conceivable that some future summer the temp will hit 40.7 C, but the temperature is not likely to shoot suddenly to 10 000 C. It has a certain level of predictability.
Above we have two examples of a system which is chaotic, and still predictable in the average: the LM and the temperature in Chicago.
A signature of chaos is that if you change initial conditions, even slightly, the system behaviour completely changes. Lets change the 'radiative forcing' to correspond January conditions. Now the temperature in Chicago varies roughly from -32.8 C to
+19.4 C. Then return the forcing back to July level: the temperature returns back to +7.2 C ... +40.6 C range! It does not wander into some completely inpredictable regime. Even though initial conditions are very different from the previous July! Even radiative forcing is likely to be slightly different (sunspot configuration is different this year, etc.).
It looks like the climate, meaning now the monthly average temperature, is robust against the initial condition of 'radiative forcing'. This seems to be true even though the *daily* weather in Chicago *is* chaotic!
One may now ask is the weather robust against other changes in initial conditions of the atmosphere, such as the CO2 concentration? I cannot say, but it sure looks like the robustness cannot be derived from the fact that the weather is chaotic.
Even if a physical system is chaotic, COE constrains it so that it is no longer completely inprecictable. COE prevents the temperature of Chicago to shoot suddenly to 10 000 C, for example. That we can be pretty certain about.
There are other physical limitations constraining the atmosphere, such as the detailed power balance. COE limits the Chicago temperature into a narrower range than -273...+10 000 C. These claims are (marginally) more controversial.
The HadCM3 circulation model implies (insert your favorite constraint here) - rather dubious.
Whether the climatologists have gotten the model details quantitatively right (implies rather strict constraints) is very controversial.
Just like the Logistic Map has a constraint which is built in the system at a theoretical level (not only observed experimentally), there is a hierarchy of theoretical constraints governing atmospheric models. COE is the least dubious of those.
I don't want to go to the depths of climatology, let's leave that to climatologists. All I'm saying is: if a system is chaotic in short term, you cannot say that its long term averages (or other statistical measures) are also *necessarily* chaotic. They *may* be, but you'd need make additional arguments about the nature of the system.
I don't even try to explain them. That is what climatologists are trying to do, and I, too, have doubts whether their models are correct. All I tried to say is that the observed short-term inpredictability does not *necessarily* imply long-term inpredictability.
The 555 is sometimes part of one of the possible solutions to a problem, hardly ever part of any of the better solutions.
There are other ways of doing what the 555 does - equally reliable and well-known though they haven't been around as long - and a recycler who couldn't come up with something better than a 555 wouuld seem to have missed out on the last thirty years of electronic development.
You are the only one to report that particular problem. Granting your obvious cognitive defects, you probably ought to talk to your physician about it.
It can be, particularly when the new problem being solved doesn't look much like the one from which you are recycling the known solution.
It's called mass production. Happily, the manufacturers of integrated circuits keep on producing new devices that can sometimes solve old problems in new ways.
Not that I've noticed.
You can tell where it is really working - half-drunk cups of cold coffee imply that the discussants are really interested in wha they are talking about.
Good luck. Can you still get cheap step-recovery diodes good enough to generate a fast edge?
And where would we find the definitions that allow us to distinguish between "weak" and "strong" non-linearity? I suspect that difference between the weather and the solar system is that the planets are massive and spend most of their time a long way apart, while the atmosphere is thin and confined to the surface of the earth.
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Probably because there isn't enough positve feedback. There's certainly enough to amplify the relatively small Milankovitch effect into ice ages and inter-glacials. Archer's book points out that what it takes to start an ice age is for the northern hemisphere summer sunshine intensity to drop to 450 W/m^2. At that point, last winter's snow survives all through summer, reflecting more sunlight than bare or forested ground, and the northern hemisphere ice sheets start to form again.
They last until the earth's orbit gets into a state where the nothern hemisphere summer gets warm enough to start melting the ice (it can peak up to 530W/m^2). It takes a while for the ice sheet to melt and for the ice to slide off into the oceans. but not nearly as long as it would take to melt the ice in place. Apparently chunks of ice slid off roughly every eight to ten thousand years during the most recent age age (Heinrich events)
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and nobody really knows why. The Heinrich events always took place in the middle of an (equally mysterious) Dansgaard-Oeschger cycle, which repeated rather more frequently - evey 1470 years.
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But, as it happens, the temperature in Chicago doesn't matter very much because Chicago is a pretty small area. Archer's book mentions the problem (in chapter 2) and reports that editing out the urban heat islands didn't make any perceptible difference to the global temperature, averaged right around the planet.
In fact we do understand the physics pretty well, but can't model the planet in enough detail to take full advantage of this fact
Climate models are constructed not to be chaotic, and to that extent may not be realistic, but you've yet to prove your claim that because weather is chaotic, climate has to be chaotic. Ocean currents do look as if they might be chaotic
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but we don't know much about most of them, and what we can see may be beats between slower and more regular oscillations, like the one that might be responsible for the Dansgaard-Oeschger oscillations.
All perfectly true, but you've yet to establish that the climate really is chaotic, and even if it were we wouldn't know how long it takes to go uncertain. Like I said, the ice cores record some 800,000 years of tolerably regular oscillations with a 100,000 year period between 10,000 years of inter-glacial and 90,000 years of ice ages.
There was a 50,000 year integlacial, some 400,000 years ago, which coincided with a period when the earth's orbit around the sun was at its least elliptical, as it is again now.
It could just be a system with some positive feedback, enough to amplify the Milankovitch effects but not enough to make it unstable, and not chaotic at all.
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Since you have virtually no experience with 555's, I don't understand
how you could make a statement like that and expect to be believed.
But, in any case, maybe you've got a trick up your sleeve and can post
an example or two of something that's better,smaller,fater, cheaper?
The gravitational effects of planets on one another is very small. If Jupiter disappeared, the effect on Earth's orbit would be subtle, and take a long time to accumulate into noticable effects. And Earth's orbit wouldn't change much, ever. In a strongly nonlinear system, like a close-orbit 3-body system (say, a single star close to a binary) a small change to the state of one star would, in not too many orbits, grossly change the configuration, maybe even eject one of the bodies.
The definition is circular: if a small change to the state of the system will, before too long, produce a gross change to the state, it's strongly nonlinear. Lots of physical systems are strongly nonlinear. Weather certainly is. Climate most likely is, too.
I'll lay you odds that Slowman can't even lash-up a 555 as a one-shot and get it running properly. ...Jim Thompson
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I can see November from my house :-)
If you have reports of rainfall from upwind stations, and you see a big front approaching on radar and satellite, you can usually "predict" rain. Except on the West Coast, where the front may make a hard left turn a few hours before it's expected to arrive.
And you can always average rainfall data from the past 20 years and "predict" the probabily of rain on any given day of the year. Whoopee.
What people can't do very well is acquire a set of initial conditions and plug them into an atmospheric simulation and do much better than the above. Ten days is pushing prediction. Six months (as in an upcoming hurrican season) is impossible. In fact, the models seem to be wronger than chance, agressively wrong.
And none of the nodes of an opamp-based chaotic oscillator will ever go beyond the supply rails. Calling that "predictability" is silly.
The constraint is not as hard
Yes, but the difference between 7 and 40C *matters*. As does the difference between a warm period and having most of North America covered by glaciers.
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