I don't know enough about Fractals (Question!)

Is there a way to determine chaos exists, as in a self-referential chaos, without knowing the equation behind it?

My guess is no. No matter how far you scan what appears to be a fractal, you'll never know if it's truly chaotic without knowing the fueling equation because it's always possible it will repeat.

But what methods of scanning for chaos are there, and is there a way to determine a relative degree of certainty that a system is possibly chaotic?

I ask because it has ramifications when scanning a system of axioms for self-replicacy (at the very least, in a finite number of steps). Say, for example, trying to determine a general set of rules for scanning a primordial-soup simulator for DNA-analogues.

I'm not sure what you mean. What do you mean by chaos? Do you mean the lack of repetition? Many chaotic systems are extremely repetitive but with slight variations; do you mean exact repetition? If you tessellate the Mandelbrot set you have exact repetition but it's just as chaotic.

With physical systems it's worth bearing in mind they don't have infinite fidelity.

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Originally Posted by Xei

I'm not sure what you mean. What do you mean by chaos? Do you mean the lack of repetition? Many chaotic systems are extremely repetitive but with slight variations; do you mean exact repetition? If you tessellate the Mandelbrot set you have exact repetition but it's just as chaotic.

With physical systems it's worth bearing in mind they don't have infinite fidelity.

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I mean lack of repetition while adhering to rules of self-reference. As in fractals.

I'm pretending the system does (or any system--I get to decide which axioms go toward lengthening/shortening a 'chemical' chain. Think A New Kind of Science-type programs (though I've yet to read the book)). I don't care about fidelity, I care about the ability to use a program to scan for self-replicability as it could possibly emerge from self reference.

Fractals are defined by the PRESENCE of repetition. The repetition must at least be approximate and is often exact. Please don't get all terse when you're not using words in standard ways.

I really can't say any more without concrete examples of what self-replicability and self-reference mean in context. What you're saying is too vague for a mathematical answer.

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Originally Posted by Xei

Fractals are defined by the PRESENCE of repetition. The repetition must at least be approximate and is often exact. Please don't get all terse when you're not using words in standard ways.

I really can't say any more without concrete examples of what self-replicability and self-reference mean in context. What you're saying is too vague for a mathematical answer.

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I need more words then. :x

I'll try asking this one:

Fractals are the presence of repetition, but each copy is not an exact replica, just self-similar. Right? And there's no discrete way to say that a mandelbrot has been zoomed to one full "level," because there are no discreet levels.

So if program A was your standard zoom program like this, could there exist a program B, whose input is the fractal image itself (and whatever relative zoom the program asks for) determine it was a true fractal, as opposed to a repeating pattern, or something truly chaotic? The existence of program B implies that this could always be done in a finite number of steps (If not always, then is there a "sometimes"?).

What do you mean there is no discreet levels?
As in there are no defined levels? No objective levels? Just what people choose?

Afaik, the next "level" is the next sequence in the algorithm.

http://upload.wikimedia.org/wikipedi...ral_34.svg.png

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Originally Posted by Abra

I'll try asking this one:

Fractals are the presence of repetition, but each copy is not an exact replica, just self-similar. Right?

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Wrong. Many fractals, like IFS (iterated function systems), are constructed based on the precise idea that they contain exact copies of themselves. Example: seirpinski gasket click me

I think you're also very confused as to what the mathematical term "chaos" means. I won't give the exact definition, but chaos is a property of dynamical systems whereby the input values are 'mixing'; you can get arbitrarily large differences in outputs from arbitrarily small differences in inputs. A priori, this has NOTHING to do with fractals. The only connection between fractals and chaos is the fact that some dynamical systems are both fractal and chaotic, like Mandelbrot*.

*I actually mean the quadratic family of dynamic systems is chaotic, and by applying an escape algorithm on the complex plane, you can generate the Mandelbrot set, which is a fractal on the plane

So I guess there could be discreet levels, then. They'd be indistinguishable from any other level. Right?

In that case, in this entire thread I mean to talk only about dynamical systems like Mandelbrot (that are fractal and chaotic--though what are examples of chaotic dynamic systems that are not fractals? Is that like how emergent properties of chemistry, as chemicals grow larger and more complicated, give rise to properties of biology?). Ones that are chaotic and do not produce exact replicas.

In a dynamical system, there are technically no discreet levels, as there would be for a sierpinski. Is this true? Also, I repeat my question about program B a few posts above, this time only as it applies to dynamical system fractals.

On the side, what do you mean by applying an escape algorithm? What is that?

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Originally Posted by Abra

So I guess there could be discreet levels, then. They'd be indistinguishable from any other level. Right?

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Depends on what kind of fractal we're talking about. All fractals have self-similarity but only a subset of those have exact copies of themselves.

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Originally Posted by Abra

In that case, in this entire thread I mean to talk only about dynamical systems like Mandelbrot (that are fractal and chaotic--though what are examples of chaotic dynamic systems that are not fractals? Is that like how emergent properties of chemistry, as chemicals grow larger and more complicated, give rise to properties of biology?). Ones that are chaotic and do not produce exact replicas.

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The Mandelbrot set is a set of points in the complex plane defined by whether or not a given point escapes to infinity when the complex quadratic function is applied to it repeatedly. It's a set, not a dynamical system. The dynamical system is the quadratic function.

Fractals are sets. Dynamical systems are functions. You may as well be asking whether there's such thing as a grapefruit that can fly. Now, fractals can be generated by drawing the results of dynamical systems in certain ways, but you would need to define that. I don't think anyone can tell you definitively whether or not there's such thing as a chaotic dynamical system that can't be iterated in such a way as to produce a fractal set. Aside from maybe the trivial systems.

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Originally Posted by Abra

In a dynamical system, there are technically no discreet levels, as there would be for a sierpinski. Is this true?

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See above.

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Originally Posted by Abra

Also, I repeat my question about program B a few posts above, this time only as it applies to dynamical system fractals.

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Ok, so you want to know if there's a program that takes as input a set of points and determines whether or not it's a fractal in a finite time. My instinct says the answer is No, because a fractal can be arbitrarily complicated and memory space is finite. With infinite memory, maybe. That's a deceptively difficult thing to answer, because it involves varying degrees of "infinity". You would probably have to ask a PhD to get a straight answer.

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Originally Posted by Abra

On the side, what do you mean by applying an escape algorithm? What is that?

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Pick a point in C (the complex plane). Apply the dynamical system z -> z^2 + c (where c is some constant to determine the location of the set you end up drawing), and do it X number of times until |z| goes to infinity ("escapes" to infinity, hence the name) or you're satisfied that it won't go to infinity. If it escapes, it's not in the Mandelbrot set. Otherwise, it is.

Example:

z = 1 + i, c = 0.

(1+i)^2 = 2 + 2i, so |z| = sqrt(8)
(2+2i)^2 = 8 + 8i, so |z| = sqrt(128)
etc...

You can see that |z| grows very quickly. Obviously, (1+i) is not in the Mandelbrot set. You can also perhaps see why the Mandelbrot wasn't discovered until the 1970s, when computers became common.

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Originally Posted by cmind

Fractals are sets. Dynamical systems are functions. You may as well be asking whether there's such thing as a grapefruit that can fly. Now, fractals can be generated by drawing the results of dynamical systems in certain ways, but you would need to define that. I don't think anyone can tell you definitively whether or not there's such thing as a chaotic dynamical system that can't be iterated in such a way as to produce a fractal set. Aside from maybe the trivial systems.

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But 'iterative/simulated' chemistry can be described as a linear combination of functions as defined by physics, most of which decay exponentially (and biology, which I consider the starting point at which simulated chemistry produces a self-replicable molecule). So what kind of set of functions qualifies as a dynamical system, then?

I'm not asking if any dynamic system can turn into a fractal. I asked a question about dynamic system fractals, because the answer would imply something similar about dynamic systems. If iterative chemistry is a dynamic system. Which I think it is, but you'll need to give further definition. I see a similarity but I'm not sure if I'm hitting the right terms.

Given a set of axioms (an incomplete one) to simulate iterative chemistry (just seeing how molecules grow per iteration, starting from elements), if I was trying to scan for multiple compounds that are self-replicable (DNA and DNA analogues), would I be able to do this always (I say no), sometimes, or never in a finite number of steps?

Bump (I hope the above post gets addressed this year!). Also requesting any reading that might help someone new to dynamic systems.

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(and biology, which I consider the starting point at which simulated chemistry produces a self-replicable molecule)

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Tsk tsk, Abra. Also has to be a molecule with the capacity to mutate while maintaining its self-replicable property.

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(just seeing how molecules grow per iteration, starting from elements)

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Starting from the elements, and acted upon by an energy distribution.