Sorry I've dropped off a bit, I was away at a conference. Lets see if I can address these points.
orbits with rationally related periods are NOT tori, they are periodic orbits. and they remain in the "chaotic sea" after the tori are gone. thus the destruction of the tori says nothing about the existance of the periodic orbits. no?
1. It is completely legitimate to talk about the period of a KAM torus. It is essentially a shorthand for the winding numbers of orbits around the torus(technically winding numbers are
frequencies around the torus). We can think of the periodic orbits you are talking about as winding around either the inner or outer edge of the torus.
2. Periodic orbits are trivially destroyed in a system under small perturbation. Thus they do not exist.
3. Maybe I misunderstand, but I think you might be making a little logical mistake here, from the standpoint of argumentation. You claim that there exist chaotic systems with periodic orbits. I do not dispute this. The existence of chaotic systems with periodic orbits does not disprove my claim that chaotic systems without periodic orbits exist. To disprove my claim you need to show me that no chaotic systems without periodic orbits can exist(in the sub-discussion, that no KAM systems without periodic orbits can exist).
4. To make my claim I only need to show the existence of one system that has no periodic orbits. Here are 3 KAM systems and one non KAM system.
a. A KAM system that is defined over only quasi-periodic orbits(as is typical, because periodic orbits are not interesting in the study of the KAM theorem).
b. A system under small perturbation. Under such a system only quasi-periodic orbits survive, thus this is a system without periodic orbits.
c. A KAM torus with irrationally valued inner and outer diameters.
d. Rule 30
to the extent that we are discussing chaotic mathematical systems (to the extent that KAM is relelevant), it is important not to confuse physical systems and mathematical systems.
if you really want to talk about natural systems, you'll find it hard to establish that they are "ultimately" described by mathematics at all. many of us believe them to be, but that is a religous belief not one based on evidence.
In this topic of discussion we
are applying these definitions to natural systems. But it might be more fruitful to talk about systems that dissipate quickly and slowly(orbits of planets vs the friction of snooker balls).
i do not see how that could follow, unless you are suggesting that dissipative systems cannot be chaotic: the most commonly discussed chaotic systems are dissipative (Lorenz's equations, all systems with strange attractors, or any attractor for that matter!)
did i miss something here? what was the "this" in "this seems to exclude"?
At the limit, dissipative systems will fall into a steady state, thus ceasing to be chaotic. Thus, at best, the system will be chaotic within some finite boundary. You can think of it as being a point made in analogy to your argument against computers being chaotic from finite memory. Which is what initiated this discussion. All physical systems, digital or analog face physical limitations. Thus, it is incorrect to exclude one computational modality while including the other.
Here are your quotes arguing we should consider a system only at the limits:
it means digital computers cannot simulate chaotic processes in the long run (as all trajectories eventually fall onto digitally-periodic orbits of finite length).
the hallmarks of chaos are defined in the limit as time goes to infinity; transient behavour is almost always explicitly excluded. (not sure if i need that "almost")
Also, In the Lorenz Equations, you'll find that they dissipate quite quickly if you include a term to allow dissipation.
a little hasty, but not much. a dense set of unstable periodic orbits often features in the various definitions of choas, not just bob devaney's. and some argue that the existance of such a set follows from other definitions of chaos.
Some might argue this, but to claim that this is proven constitutes conjecture. Moreover, some argue the contrary and I've not seen any clear explanation why a dense set of periodic orbits must follow from chaos.
yes, i think that is the main point. no hamiltonian counter example has been provided at this point.
Who cares if a system is Hamiltonian? A non-Hamiltonian example would suffice and I don't think you've presented
any objection to Rule 30.
there is no evidence for this in an analogue system, the fact that the observation of the state is quantized by the A/D converter implies we have a limited number of values we can observe, not that the system has a limited number of states.
Now you're arguing that
no physical system can be digital. If that is your opinion, then this whole discussion is really superfluous. Personally, I think that we should characterize a physical system by its observables and not by the possibility of an underlying unobserved state. It just feels...
unscientific.
But just to avoid blurring the line between the physical and the theoretical, I can't see any reason why a theoretical turing machine with an infinite tape can't constitute a chaotic system.
i know of no cases in the lab where this has been a problem. there are other problems, of course; and i am not sure what you mean by "controlable": you can certainly damp out chaos if you try, but if you do not actively attempt to, then the observations tend to look as if they came from a chaotic process.
There is a whole field of engineering called "Control Systems" that studies controllable systems. These are physical systems that can be made to perform reliably. If you are doing some sort of analog simulation it is important that the system be controllable, at least in the sense that you can make it perform like the system you are simulating and not just like itself.
Noise induced stability, however, is a common phenomenon in chaotic systems. This can cause a chaotic system to perform regularly under perturbation. If you are unfamiliar with this, I'm sure you can find it in the literature. I think I even linked to a tutorial describing it in one of my previous posts.
Whether stability is induced or not, an analog simulation will reach a floor(often a thermal noise) where it is forced to deviate from the trajectory of the system it is simulating. At that point it is just "shadowing" the original system as well.(Insofar as it is following a path in the chaotic system, but not the original one)
agreed, but the point is that that path is not chaotic: the periodic computer trajectory is shawdowed by an unstable periodic orbit of the chaotic system.
As far as I'm aware, the shadowed trajectory does not need to be periodic, at least not theoretically. It seems you are restating the point from the finiteness of computer memory
how does filtering the observations have any effect on the dynamics of the system?
I'm talking about filtering the system itself, not just filtering the observations. So how can it not have an effect on the dynamics of the system?
