Chaos is a complicated topic. There are lots of preliminaries that are useful to understand if you really want to grasp what chaos is, and how you can usefully analyze and characterize it. One of the really useful concepts is attractors. Attractors aren't specific to chaotic systems - most real-world dynamical systems have attractors. But attractors are extremely useful in understanding chaotic systems, and how they different from more tractable dynamical systems.
So what's an attractor? Take a dynamical system we described in the last post, and look at its phase space. The phase space isn't just a random cluster of points. It's got a very rich structure to it. An attractor is a piece of structure of many systems that appears when you view the systems evolution over time. Speaking informally, an attractor is sort of like a black hole in the phase space: it's a region of the space where if you get sucked into it, you'll never leave.
Attractors have shapes, and the shape of the attractor can tell you lots of interesting things about the behavior of the system. One interesting bit of trivia to convince you to read the rest of this: chaotic systems have attractors, and the attractors for chaotic systems usually have fractal shapes.
Ok, so let's get formal, so that we can precisely define what an attractor is.
To start, we can adopt a notation for talking about a dynamical system that makes the definition of the attractor easier. Suppose you have a dynamical system with phase space P (to review, that means that every possible state of the system is a point in P). You can describe P using a two-parameter function, f(x,t), where f : P×ℜ→P, where x is a point in P, t is a unit of time, and f(x,t) is another point in P. The value of f(x,t) is the state of the system after the time interval t passed since it was in state x. That's a bit of a mouthful, but the basic concept isn't difficult. A dynamical system is a system whose behavior is parametric on time. f, which is called the evolution function of the system, basically says "If you run the system for t seconds starting from state x, you'll wind up in state f(x,t)". So f is a complete description of behavior of the system, which maps points in the phase space to points in the phase space after some time has elapsed.
One important thing to understand is that the function f exists in theory, but for many dynamical systems, we can't find out what that function is. If we've got a system that's described using differential equations, we might be able to integrate them analytically, and figure out what f is; but for most systems (remember the old rule: most of everything is bad), we either don't know how to analytically integrate them, or we know that we can't analytically integrate them. So the best we can do in practice is to construct iterative approximations of the value of f(x,t). But in theory, there's always a function that describes the system, whether we can get our grubby hands on it or not.
So, back to the attractors.
(The image to the right was created by Nathan Selikoff. You can view the full-size version of the image, and Nathan's other work here.
For many phase spaces, there's an interesting property, which I call the black hole (that's not a standard term to the best of my knowledge; but it seems apt to me.). For some subset of A⊆P, if p∈A, then for all t≥0, f(p,t) ∈A. In english, there's some region of the phase space where once the system's state enters that region, it will never leave it. It can cycle around forever, never crossing through the same point twice, but it will never leave the region. The black hole is the attractor.
I've heard all too many people quote just that statement above as the definition of an attractor. Just like people try to talk about chaos by just citing the sensitivity to initial conditions, and ignoring the more complicated (and frankly, more interesting) topological phase-space properties, people talk about attractors while leaving out the more complicated and interesting parts.
In addition to the black hole property, to be an attractor, the black-hole region of the phase space needs something called a basin. The basin is another subset of P. If the state of the dynamical system ever enters B, then at some point in time, it will come arbitrarily close to A - that is, if b∈B, then limt→∞f(b,t)∈A. So in addition to the trap region, if you get too close to the black hole, then while you might never actually fall in, you're still trapped; you'll eventually get pulled closer and closer to the attractor. You might be able to get incredibly far away for some period of time, but your eventual fate is sealed: you're going to fall towards it.
So - an attractor is a region of the phase space of a dynamical system which has a black hole and a basin. The final property that you need to complete the definition is that the attractor is the smallest subset of the phase space P that has those properties - that is, it's the smallest black-hole region with an attractor basin.
To give you a simple example of an attractor, think of Newton's method for finding the root of a polynomial. The black hole is the fixed point of the the calculation - the set of true roots of the polynomial. The attractor basin is the set of points for which Newton's method will converge towards the fixed point. In real computation, for most polynomials, you'll never get all the way to the exact root, but if you keep iterating (that is, you keep stepping forward the time), you can get as close as you want.
Finally, aside from the mathematical utility, many attractors are just beautiful to look at. Through this post, I've included images of attractors.
Physical Science
Comments
I have a few questions. If the system enters B it is possible (but not guaranteed) for it to enter A in a finite amount of time, right? A has to be a proper subset of B, right? Can you post some links to larger versions of the pictures in the post?
Posted by: Jonathan L | June 26, 2009 12:51 PM
emphasis added in the first
These two contradict each other. As your later example shows, fixed points and periodic orbits can very well be attractors. They're just not chaotic attractors.
Posted by: John Armstrong | June 26, 2009 1:12 PM
Nice to know what is really meant by "chaotic attractors" when it's used (like many other math/quantum terms, without any real meaning) in many sci-fi shows today. :)
Posted by: Joe Shelby | June 26, 2009 1:53 PM
Like with your previous post on chaos, I now understand a lot of concepts very clearly - the black hole, the basin - but I don't understand why they are necessary. Why does a chaotic system need attractors and why do attractors need a basin?
Maybe this will become clear in the next posts, in which case I have said nothing. ;)
Posted by: Martijn | June 26, 2009 2:13 PM
Re #2:
They don't contradict. Let me change the stress a bit: It *can* circle around forever, never hitting the same point twice, but it will never leave".
It's definitely not one of my clearer bits of writing. My intention was to try to say that it's *not* necessary for something to endlessly repeat itself once it's caught by the attractor - it can have a continually varying, never repeating path. The key constraint is that it can't escape. But it does, unfortunately, have the reasonable reading that I mean that that's a *requirement*, when I meant it as a *possibility*.
Posted by: Mark C. Chu-Carroll | June 26, 2009 2:35 PM
@John Armstrong:
For the fixed point of the phase space, it is never touched by an orbit other than the one that stays forever in that point. It is an attractor but any orbit will reach it at t=+∞ (that is not a real point: it will lower its distance from the attractor but never reach it: this is the asimptotycally stability concept).
For the periodic orbits, the point is intended in a more complete point of view: different solutions can never pass from the same point (t, x1, x2, .., xn) for the Picard-Lindelof theorem. At different times, they theoretically can. Some systems, the autonomous ones, have trajectories that are always the same WHENever you start (think of a ball that falls down a hill: its path does not depend on the hour of your watch when you push it). The trajectories in the phase space are the projections of the ones on the real spaces, that has also time as a coordinate; they are the same for every chosen t0.
If two solutions of an autonomous system cross the same point on the phase space, we can traslate one of the solution in time to encounter the other and the theorem will be violated. If the ball goes down the hill and steer left at a certain point, it will always do this. Note that here speed of the ball is a dimension in the phase space.
Posted by: Giorgio Sironi | June 26, 2009 2:41 PM
These are fun, and I like the addition of the set theoretic info. Perhaps you could include an example of an explicit chaotic system, in Xdot(t)=f(X(t)) form? That would be cool.
Posted by: AnyEdge | June 26, 2009 4:07 PM
I never had a chance to study chaos theory, but I do love the beautiful fractals that are generated from these studies.
One question - are systems such as these used to study quantum physical environments? Seems to be a great tool for this purpose.
Posted by: steve | June 26, 2009 6:12 PM
Re Martijn (No 4),
I don't believe that all chaotic systems do have attractors---merely that they are useful and interesting when they do. Some non-chaotic systems have attractors too, consider the system given by f(x, t) = x/2^t (for t in the natural numbers, this is just equivalent to dividing by 2 over and over). x=0 is a fixed point, and eventually all real x tend towards zero, so it has a basin, but this is not chaotic in any sense of the word.
Attractors need a basin, however, because otherwise they're not terribly interesting. In fact, the name comes from the fact that they "attract" orbits to them. Chaotic systems may have closed subsets without attracting basins, but then for the points not in the closed subset, the "black hole" might as well not be there at all---it's irrelevant to the rest of the system.
Lastly, without the basin requirement, A=P would satisfy the definition---any point in P has an orbit that stays in P forevermore. That's trivial and boring though.
Posted by: MPL | June 26, 2009 7:09 PM
This is an interesting post, with beautiful images. However I am a little worried by your lack of attribution for the images. I recognise one at least as the work of Nathan Selikoff: http://www.nathanselikoff.com/strangeattractors/#aesthetic_explorations
Posted by: Edmund Harriss | June 27, 2009 1:45 AM
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Posted by: kristina | June 27, 2009 6:35 AM
Nice post, holmes!
Posted by: Comrade PhysioProf | June 27, 2009 12:49 PM
Just beautiful and interesting MarkCC. I really enjoy your blog.
Posted by: Gabriel Girón | June 27, 2009 3:53 PM
Did I understood right that in order to make Newton's method converge, your initial guess needs to belong to the attractor basin of the function?
That's interesting since I've never seen a definite conditions to make it converge, only a (possibly incomplete) bunch of cases. Finding the attractor basin might be hard if not impossible though, but at least it's well defined.
Posted by: Matt | June 28, 2009 5:04 AM
Re #14:
Yes, it's correct that Newton's method converges only if the starting point is in the attractor basin. But all that's doing is giving a name to the convergence region. It doesn't do anything to really define the region where it converges. Finding the attractor basin is no easier than finding the convergence region if you didn't know about attractors and attractor basins.
Posted by: Mark C. Chu-Carroll
| June 28, 2009 10:49 AM
If the evolution function of a phase space is nice and smooth, does the divergence of it tell you anything interesting about the attractor?
Posted by: Paul Murray | June 28, 2009 10:57 PM
@15
From my memory of playing with "fractint", the pattern of attractor basins for newton's method is itself a fractal. When you use newton's method to get the three cube roots of 1, you can prove that between any two basins for any two of the roots, there must be a basin between the two that converges on the third.
Posted by: Paul Murray | June 29, 2009 12:15 AM
Hi Marc,
I always enjoy reading your blog.
Have a doubt (off topic). There is a Financial advisor who is saying the following:
"So I have my own way of determining the seriousness of any new threat to human life, it's called the Celebrity Death Test, and I hope you find it useful. The way it works is simple, if a Celebrity dies from the Threat then it is to be taken Seriously, if they don't then it's probably nothing to worry about. Bird 'Flu, fine. AIDS, not fine. I can remember when Rock Hudson died, that was the moment when AIDS became real for me. (I'm also a fan of Doris Day, read into that what you will!) You see how it works? It's just a statistics thing. If a Celeb suffers from it (and assuming it's not something that has a natural correlation with Celebrity or is self inflicted) then it is statistically significant for the rest of us."
http://www.wilmott.com/blogs/paul/index.cfm/2009/4/26/Celebrity-Death-Test
Lacking the math expertise, I wonder if such a thing makes any sense. I feel that he's wrong, the main attribute of celebrities is to be just a few that appear on TV. There is nothing related to taking a representative sample of, say, the whole population.
Thank you
Posted by: Fernando | July 2, 2009 8:56 AM
I don't have anything really smart to say, but I really enjoyed your chaos posts so far.
I hope there are more to come :)
Posted by: Shai Deshe | July 2, 2009 10:22 PM
Hello Mark .... I have read Mandlebrot's more simplified book on the subject of fractal finance. I know traders make use of Fibonacci numbers in analysing chart patterns. I was wondering if you think Fibnonacci series can act as some kind of 'attractor' in the financial context ?
Posted by: Celal
| July 4, 2009 7:26 PM
Is it possible that the basin is equal to the attractor?
Say for the logistic map with parameter equal to 4. Would we consider [0,1] to be a chaotic attractor? It has the black hole property, but I don't think anything is getting
"attracted" since you either start in [0,1] and bounce around in the black hole, or you start outside of [0,1] and diverge from the black hole.
Posted by: Mark | July 13, 2009 9:44 AM