Questions about chaos theory

[Genecks]

I have a couple of questions about chaos theory. I'll start with one.

 

In chaos theory, as time passes from t=0, does a system become more chaotic, thus more difficult to predict?

[Mitch Bass]

WHen you wrote "as time passes from t=0"....is this meaning the same thing as saying "as things occur from a point in the past to the future than well...regardless the simple answer to your question is that if a system does become more chaotic than it will be more difficult to predict. This being said let us examine how you phrased your question keeping in mind that I am not attempting a conflict based on semantics. You wrote "In chaos theory, as time passes from t=0, does a system become more chaotic, thus more difficult to predict?" Breaking this down we have to begin with In chaos theory, as time passes from t=0, does a system become more chaotic, thus more difficult to predict? If you are suggesting that chaos theory causes systems to become more chaotic then already a problem becomes evident. Many systems as time "passes from t=0" become less chaotic. The pathway towards user friendly computers is an example of how systems can go from greater to lesser chaos. Easier to operate, understand. Computers that were unreliable and often unpredictable evetually evolved to the point where a person can use a computer now without ever learning a computer language or even an instruction book.

 

The essence of chaos theory constitutes the concept that greater amounts of variables generates a system that is not random but rather chaotic.

 

So, my question to anyone reading this, should you choose to attempt to answer is: How is chaos differentiated from randomness?

[ajb]

[/font][/color]should you choose to attempt to answer is: How is chaos differentiated from randomness?

Chaotic and random processes are not synonymous.

 

Chaos theory studies dynamical systems that are highly sensitive to their initial conditions. This includes systems that are fully deterministic and in no way random. It is not any randomness that makes the behaviour of such systems difficult to predict it is their sensitivity to these initial conditions that is the problem. You can start a system off at very close points to end up with very different trajectories.

 

In practice it may be difficult given some data to decide if the system behind this is random or chaotic, but this is another question.

 

I don't know how to answer the opening question. I am not sure what people use to measure the 'degree of chaos'.

Edited October 8, 2014 by ajb

[studiot]

 

In chaos theory, as time passes from t=0, does a system become more chaotic, thus more difficult to predict?

 

 

I suppose you could say it depends since 'chaos' is a collective term for a number of different effects.

 

Some systems are constrained so that the chaos cannot 'grow'.

Example of this would be the path of a metal ball hung above four magnets.

In chemistry the Belousov-Zhabotinskii reaction.

 

In other systems the 'chaos' can grow without limit even to the destruction of the system.

Examples of this would be Euler instability of motion.

Audio or video feedback.

 

mitch

So, my question to anyone reading this, should you choose to attempt to answer is: How is chaos differentiated from randomness?

 

ajb

Chaotic and random processes are not synonymous.

 

Chaos theory studies dynamical systems that are highly sensitive to their initial conditions. This includes systems that are fully deterministic and in no way random. It is not any randomness that makes the behaviour of such systems difficult to predict it is their sensitivity to these initial conditions that is the problem. You can start a system off at very close points to end up with very different trajectories.

 

In practice it may be difficult given some data to decide if the system behind this is random or chaotic, but this is another question.

 

I don't know how to answer the opening question. I am not sure what people use to measure the 'degree of chaos'.

 

 

I agree with ajb that chaos and randomness are different phenomena.

But also note that a small amount of initial randomness can lead to signification variation of system future history (called trajectory).

There may, as ajb says, be entirely predictable equations that the trajectory follows. Any the chaos arises from (small) random variations in initial conditions.

The course of this trajectory will depend upon the system as I said, but the instability that allows chaos is to grow or not, is inherent in the system, not the process.

Euler instabilty is one such example.

So called 'frequency doubling chaos', on the other hand contains the seeds of its own expansion in the process.

Feedback comes to mind here.

 

I expect this thread will develop further and I would be happy to expand on any of these points.

 

I would be wary of using the term degree of chaos to measure it since fractals are often included in the basket of effects and the the term degree could be confused with the Hausordf Dimension, responsible for this phenomenon.

[imatfaal]

I seem to remember one text on chaos theory that I read which did have an arbitrary but well argued delineation - they then tried to predict when (after how many iterations) iterated equations of different sorts would reach this limit (and I seem to recall found that the prediction was just as weighted on initial conditions); but apart from the number 3.7 I can remember nothing more about it.

 

Agree with everything ajb and studiot have written above - but would note that my favourite system with sensitivity to initial conditions (your guess in this case) is the newtonian iteration to the roots of a quartic. I wonder if the great man had any inkling of the chaotic nature - I find it hard to believe that he didn't but then my respect for him borders on hero-worship

[Genecks]

My question has to do with the legal system and the court of law. I've seem to have noticed something odd about law and science. I don't know how much law all of you know. However, there is a principle called "res judicata."

 

When res judicata is in effect, nothing prior to res judicata can be argued in court UNLESS the appeal has not expired: If the appeal can be argued, then res judicata is not effect on the appeal. However, when the appeal has expired, res judicata is in effect on the appeal (and res judicata has already been in effect on the hearing). However, I've come to the belief that the potential for evidence to over-turn that case will by necessity exist between the point of (1) res judicata and (2) the point at which another hearing occurs. As such, an individual can predict that the evidence to over-turn the case will exist between those two points.

 

However, I've been wondering if a person elongates the time between those two points, (1) the point at which res judicata goes into effect and (2) when the next hearing starts, whether or not it will be increasingly difficult to predict when and where the evidence to over-turn the case will be.

 

Yes, it's weird. This goes into the realm of metaphysics.

 

res judicata = "t=0"

 

Basically, I think I've discovered a way to prove any judge incompetent.

 

I say potential, because it goes into what I call "the culpability problem," whereby the reasonableness of the evidence as legal evidence is debateable.

Edited October 9, 2014 by Genecks

[imatfaal]

This is nothing to do with maths - and not really to do with res judicata.

 

When res judicata is in effect, nothing prior to res judicata can be argued in court UNLESS the appeal has not expired: If the appeal can be argued, then res judicata is not effect on the appeal. However, when the appeal has expired, res judicata is in effect on the appeal (and res judicata has already been in effect on the hearing). However, I've come to the belief that the potential for evidence to over-turn that case will by necessity exist between the point of (1) res judicata and (2) the point at which another hearing occurs. As such, an individual can predict that the evidence to over-turn the case will exist between those two points.

 

Your definition given of res judicata is needlessly convoluted and recondite - you would be better off with a simple dictionary definition and merely thinking of a time gap between the tribunal of fact in the first instance and any further appeal. It would seem much simpler to think of res judicata as a principle which applies after final appeal rather than over-complicate matter by considering a form of insubtantiated res judicata is in place between first judgment and possible appeal which is then reified by a denial of further appeal

 

"However, I've come to the belief that the potential for evidence to over-turn that case will by necessity exist" - this assertion cries out for calls of "Citation Needed". Why must the potential exist? What is "the potential for evidence" anyway - and how does it differ from evidence? What possible difference does the artificial time-frame between the court of first instance and the court of appeal (or various other pairs of court) have on the existence of evidence?

 

"As such, an individual can predict that the evidence to over-turn the case will exist between those two points." Much as an individual can predict the lottery numbers - either so generally that no new useful information is gained or specifically but without any possible credibility whatsoever.

 

"However, I've been wondering if a person elongates the time between those two points, (1) the point at which res judicata goes into effect and (2) when the next hearing starts, whether or not it will be increasingly difficult to predict when and where the evidence to over-turn the case will be." You claim this is metaphysical - it is not. It is baseless supposition.

 

"Basically, I think I've discovered a way to prove any judge incompetent." If you mean incompetent as unskilled - then good luck with that. If you mean you can show that court's lack competence to hear cases then you would have to address where any court's competence arises from - which is a case of massive debate which very rarely turns on matters of evidence but more on societal acceptance.

 

"I say potential, because it goes into what I call "the culpability problem," whereby the reasonableness of the evidence as legal evidence is debateable." Again I feel this sentence is more salad-like than it should be; it looks nice but has no substance

[Genecks]

If you want to throw it in another board, by all means. If it's speculation so be it.

 

I'm thinking this all falls under the mathematical universe hypothesis.

 

I'm the citation. This is original research. I've not seen anything like this before.

 

What possible difference does the artificial time-frame between the court of first instance and the court of appeal (or various other pairs of court) have on the existence of evidence?

 

 

Something I've been terming "the culpability problem." It's as though things contradict each other. There will be a contradiction.

 

I got into a legal issue a while back. I've since then been attempting to falsify the judge. I decided to use science and philosophy against the judge. In doing so, I came across a witness that discussed transcripts that contradict the judge. Interestingly, allegedly the judge marked out the transcripts. But there is a witness. Basically, no judge is perfect. No judge is their "reasonable person" ghost God. And then you have to consider entropy, so things break down.

 

What struck me as odd, however, was the proximity (the temporal location) of the contradicting evidence: between (1) res judicata and (2) the hearing. I don't think the potential exists for contradicting evidence. I think it does exist at all times. However, to have access to it, a person would have to be a judge (or perhaps court reporter). It exists in the transcripts (what was said in hearings). I came to this belief after encountering a bunch of Jungian synchronicities.

 

This gave rise to the belief that judges inevitably contradict themselves.

 

If you think I'm looney, then feel free to test my theory. I think I have a way for it to be tested.

 

It seems to me, the way to test the theory (at least best) is to falsify a judge rather than using case law with other judges. That focuses the issues down a bunch, so a person doesn't have to look at case law all over the state. That basically means showing up to all of the hearings the judge has directly after the hearing a plaintiff/defendant has. If there is closed hearing, then that would generate a problem. Eventually, the judge will generate a potential contradiction between "res judicata" and the next hearing. Unfortunately, for what I seem to have come across, judges may be aware of this conundrum, thus finding a way to omit transcripts and remove evidence.