My guess: because the abstract double pendulum is a model that only approximates real double pendulums. Analogously, computers based on Turing Machines exist despite Turing Machines requiring an infinite amount of space.
Also, a minor nitpick: the post seems to equate 'not periodic' or 'chaotic' with 'is random', which I think is misleading. Pseudo-random, I guess, but the output of a simulation of a chaotic system is highly compressible and thus not random in the information theoretic sense.
This subject is somewhat outside of my area of knowledge, but if I had to guess, this is what resolves the paradox. An abstract mathematical chaotic system is different than a real-life one, and it is my guess that if we performed the experiment, we would find the orbits would be different.
QM has withstood every test thrown at it so far, and while it's always possible that a correction to it will be found, I think it will be a very rare event should it happen.
Is that really true? Losslessly? Do you mean the entire phase space, or the sequence generated on any given run? Because I was under the impression that what distinguishes a chaotic system is that the output cannot be predicted in any way other than running the simulation (the article uses the word 'random', but it might be better to say 'unpredictable'). Compressibility relies on exploiting regularities, and regularity is predictability, isn't it?
The program that simulates the chaotic system is shorter than the unbounded amount of output being generated, and counts as a compression of the output.
I guess in other words, the more interesting question is this: can we generate hard to compress pseudorandom sequences with simple algorithms - if the algorithm description by itself can not be used as "compression"?
If you add a rule such as not using Algorithm X, it's trivial to create an Algorithm Y that generates the same output and it's barely any longer than X. For example, by adding any meaningless instruction at the start of algorithm X.
Of course, discovering what the shortest expression for a given sequence is is not a computable problem. So we can't always know if we've found the best form of compression for a sequence, we can only find an upper bound.
Or does 'simple' mean like an extractor for a standard archive file? Did you know that RAR is turing complete?
Maybe 'simple' has to be defined by limiting how much execution time you have per output byte?
You're confusing theoretical compressibility with the actual compression ratios you get when you put a string through typical compression functions.
Here's a different example. Suppose you have ASCII text encrypted with AES with the AES decryption key appended to it. Run that through gzip and you'll probably get output which is larger than the input. But here's a different compression algorithm that will work much better: Decrypt the encrypted data with the key and then compress the plaintext with gzip.
It isn't that the original string is of the sort that can't be compressed, it's that you need a compression function which is suited to the input. Pi is like that.
Compare this to other strings, like the output of an environmental hardware random number generator, which will almost always be totally incompressible whatsoever.
In randomness theory, the definition of the most compressed form of a sequence S is, given a computer C, the shortest program with output S in C.
So the program that generates the output of the simulation of the pendulums is a compression of it's output. Since it's and infinite sequence that can be compressed into a finite program, it is not random from the perspective of information theory or randomness theory.
> Classically chaotic systems generate information over time.
Really? Simply being chaotic does not preclude a system from being deterministic. Every future state of the system is "present" given the initial conditions, even if it isn't predictable.
The situation is analogous to that of the digits of irrational numbers. I can't tell you a priori what the 2^1000th digit of pi is, but if I calculated for a million years I could find it out. It's not being "generated", it's just as much a part of pi as 3.14 is, just a little harder to access.
There may be true non-determinism in nature, but it isn't necessary for a system to be considered chaotic.
The idea is that, presumably, this list of times "looks" very random and can't be expressed by a much shorter, compressed, set of data, not even by providing the pendulum equation, and initial conditions with finite precision. (You could, if it weren't chaotic, just a single pendulum.) This would mean it has high entropy content, in the information theory sense. (I don't actually know, since I haven't done the experiment nor read about it being done, but it's a reasonable guess.)
Generally it seems to me that the OP gets off into some not very well defined and not very relevant topics and, instead, for the 'chaos' he is observing there's a fairly easy explanation: The system is unstable. Or, in one step more detail, the system really is an initial value problem for an ordinary differential equation, but, going way back to Bellman's work on stability theory, it's long been well known, just from the equations, that the solution can be 'unstable', that is, small changes in the initial conditions (values) can result in large changes in the solution. And that is just from ordinary differential equations without considering quantum mechanics. And for the 'chaos' in the OP, that's about all the explanation that is needed.
Why? Because there is really no chance that the motion of the system could be periodic or even simple because it nearly never gets back accurately enough to an earlier state. So, the system often gets back to something 'close' to an earlier state, but the system is so unstable that 'close' is not close enough so that the earlier state and the present one close to that earlier state soon result in very different solutions for the future.
Something similar happens with pseudo random number generators, e.g., the usual linear congruential generators where we set
R(n + 1) = ( A * R(n) + B ) mod C
for n = 1, 2, ..., and R(1) some
positive integer. Then roughly
the R(n)/C are independent, identically
distributed, uniform on [0,1). One
of Knuth's recommendations (in one of the
volumes of TACP) was
A = 5^15, B = 1, C = 2^47. So, a
point here is, get such 'random'
numbers without considering phase
space or quantum mechanics.
In a sense, this is a very old point:
E.g., one dream before about 1900 was
that we could observe the present
state of the world and, then, use deterministic
physics to predict the future. So,
as I recall, it was E. Borel who
did a calculation and concluded that
a change of moving 1 gram of matter
1 cm, or some such, on a distant star would invalidate
predictions on earth after just milliseconds
(presumably starting after the travel
time of light from that star to the earth).
We suspect we see much the same in
weather prediction: Small changes in
initial conditions too soon make changes
large enough to switch between rain
and sunshine. The usual joke is that
a butterfly could flap its wings and
convert a clear day to a hurricane.
We anticipate that probabilistically
weather prediction is quite stable, that
is, what is stable is the conditional
probability distribution of the variables
we use to measure weather conditioned
on the present. In particular, we still
believe in the law of conservation of
Also we should notice the classic work
on ergodic theory, by Hopf, Poincare,
Birkhoff, etc.: The standard illustration is
pouring cream into coffee and stirring.
Then the theorem says that, if stir long
enough, then can make the cream separate
from the coffee back to as close as please
to the original state. Why? Because
if take the 'volume' of the possible
states at some point in time and then
let time pass, then the 'volume' of
those states is still the same. So,
as the system evolves, it is 'measure
preserving' in state space. So, if
want to apply this to a frictionless
double pendulum, then can get it to
return as close as we please to its
initial state, but between then and
now it is free to do a lot. This
stuff goes back to the first half
of the last century.
Likely there are some interesting
and important questions in chaos theory,
but what the OP is saying about the
double pendulum seems to have a simple
Am I missing something?
I guess this is relevant to the problem as well.
But thanks for the pointer.