How Are Fractals Related to Chaos?

[billiards]

I have studied fractals and think of them as things that exhibit self similarity at different characteristic scales. For example a grain of sand looks a bit like a pebble, which looks a bit like an outcrop, which looks a bit like a mountain. If you count the number of grains of sand there will be loads of them, the number of pebbles will be a bit less, the number of outcrops less still, and there will be even less mountains. I understand fractals to have a statistical distibution whereby there are lots of little ones and less big ones, and the slope of this distribution (plotted logarithmically) is related to the fractal dimension.

So how does this link to chaos?

Is it that chaotic phenomena tend to exhibit fractal statistics? Surely there's more to it than that.

Thanks for help in my understanding!

 

[billiards]

No takers on this?

Probably I'm in the wrong forum.

I've read up on chaos too. But somehow I seem to have just missed the link.
Any help would be greatly appreciated.

 

[mathman]

from wikipedia:
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Chaos theory is a field of study in mathematics, physics, and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general.[1] This happens even though these systems are deterministic, meaning that their future dynamics are fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.

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A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. Roots of mathematical interest in fractals can be traced back to the late 19th Century; however, the term "fractal" was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2]

A fractal often has the following features:[3]

It has a fine structure at arbitrarily small scales.
It is too irregular to be easily described in traditional Euclidean geometric language.
It is self-similar (at least approximately or stochastically).
It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).[4]
It has a simple and recursive definition.

 

[billiards]

Cool. So what's the link? (Or is there even a link? I presumed there must be as these subjects often appear together in books.)

 

[Claude Bile]

Chaos and fractals are described by the same type of equations, that is, when one plots or graphs chaotic behaviour, one tends to obtain fractal-like patterns.

Claude.

 

[Phyisab****]

I took a class called chaos and fractals and I am still not sure I can give really satisfactory answer. I would absolutely love to hear someone who really knows what their talking about answer this. I should ask my former professor, Chris Danforth at UVM, but I'd feel bad bothering him. One of the most impressive teachers I've ever had the pleasure of taking a class with.

One good link though is that chaotic systems can often be described as having fractal basins of attraction. In that case, no matter how exactly one knows the initial conditions of the system, you still can't tell which basin of attraction they lie in, because the basins are fractals.

 

[billiards]

Chaos and fractals are described by the same type of equations, that is, when one plots or graphs chaotic behaviour, one tends to obtain fractal-like patterns.

Claude.

OK, either you're saying that when one plots chaotic behaviour they end up with something that looks like the Mendelbrot set, or that when one plots chaotic behaviour they end up with something that obeys fractal statistic?

I'm guess you're talking about the fractal statistics.

In which case I'm trying to think of an example. Say the weather, that's a well known chaotic system, I'm thinking to take wind speed for an example. Without being a meteorologist (and without wanting to go the lengths of actually getting data for a hypothetical example), I would guess that for a typical weather station the wind is at some ambient level most of the time, occassionally you get winds picking up a bit, and even more occasionally you get quite strong breezes, and even more occasionally than that you get horribly strong gales, and once in a blue moon you get a full on hurricane/tornado type of thing. So let's imagine that this actually really is an example of a chaos (it might be, I honestly don't know) and that this really does obey fractal statistics -- then would this be an example of the link between fractals and chaos?

One good link though is that chaotic systems can often be described as having fractal basins of attraction.

What does this even mean?

 

[Phyisab****]

Which part of it didn't make sense? I thought that was a very straightforward connection.

 

[billiards]

Yeah I'm sure it is, I think you misunderstand me.

I meant what is a basin of attraction?

I didn't learn about that.

 

[Phyisab****]

From Alligood, Sauer and Yorke:

Let f be a map on R^{n} and let p be an attracting fixed point or periodic point for f. The basin of attraction of p, or just basin of p, is the set of points x such that

\left|f^{k}(x)-f^{k}(p)\right|\rightarrow 0,\ as\ k\rightarrow\infty.

In other words, the basin of attraction is the set of points whose orbits converge to an attracting fixed point or periodic point. If the basin of attraction is fractal, any orbit will be at the very least *extremely* sensitive to the initial conditions, if not (not sure I have forgotten much, could try to prove or disprove if you are feeling ambitious) chaotic.

PS what is the latex code for a nice double struck R, as in real numbers?

 

[Phyisab****]

The chaos theory I learned was written in this kind of language, it may help you understand although it does not explicitly mention basins of attraction.
http://en.wikipedia.org/wiki/Stability_theory

 

[Phyisab****]

I want to hear a serious answer to this question by an expert.