pollack@dendrite.cis.ohio-state.edu (Jordan B Pollack) (06/26/91)
Interestingly, Herman Rubin says of the (integer) hailstone sequences: >>The chaotic nature is that a slight difference at one point >>makes a big difference a considerable time in the future. Most of the widely discussed "chaotic" dynamical systems, like 4x(1-x), iterate over real numbers. But real numbers are not computable, and fractal images (presented by their proponents) are always floating point approximations to a platonic ideal. I find the "chaos" idea, of an iterative system having a "deterministic aperiodicity", quite stimulating precisely because it does not seem computable. I played with the hailstone numbers for a couple of hours once, and found them somewhat "unpredictable". It seemed to me that the sequences are the result of a non linear dynamical system over the integers (x -> x/2 if even, x -> 3x+1 if odd). But the hailstone system, though sensitive to initial parameter, always seems to halt by finding the (1-4-2-1) attractor, though I'm not sure this has been proven. Other non-linear operations, like modular multiplication by a constant, (the basis for pseudo-randomness), also lead to simple periodic cycles. Are there any deterministic functions over the integers which are aperiodic? Such a function would have to traverse an infinite subset of the integers in an "unpredictable" way, without halting or cycling. "Get the next prime number" would seem to be a candidate but for its monotonicity...
aboulang@bbn.com (Albert Boulanger) (06/30/91)
In article <POLLACK.91Jun25121703@dendrite.cis.ohio-state.edu> pollack@dendrite.cis.ohio-state.edu (Jordan B Pollack) writes:
Are there any deterministic functions over the integers which are
aperiodic? Such a function would have to traverse an infinite subset
of the integers in an "unpredictable" way, without halting or cycling.
"Get the next prime number" would seem to be a candidate but for its
monotonicity...
Interesting question, Jordan. Such a dynamical system whould have to
use large integers like normal chaotic ones use the fact that there is
"plenty of room at the bottom" to do its folding. One of the criteria
for chaos is ergodicity or that the orbits have to be dense. You seem
to have a conflicting desire to have it spread out as much as
possible. Anyway a good book on integer dynamical systems is:
Discrete Iterations
Francois Robert
Springer Verlag, 1986
Also there is an intersting n^.5 scaling of the size transient path
before these systems fall into their terminal cycle or fixed point.
For more information on this, see:
"The Simulation of Random Processes on Digital Computers: Unavoidable
Order" T. Erber & T Rynne, J of Comp Phys, 49, 394-419 (1983)
For possible quantum mechanical implications, see:
"Randomness in Quantum Mechanics -- Nature's Ultimate Cryptogram?"
T. Erber & S. Putterman, Nature, Vol 318 (7 Nov), 41-43, (1985)
Shifting my bits using a Mixmaster (TM),
Albert Boulanger
aboulanger@bbn.com