More about Dense Periodic Orbits

Based on a recommendation from a commenter, I’ve gotten another book on Chaos theory, and it’s frankly vastly better than the two I was using before.

Anyway, I want to first return to dense periodic orbits in chaotic systems, which is what I discussed in the previous chaos theory post. There’s a glaring hole in that post. I didn’t so much get it wrong as I did miss the fundamental point.

If you recall, the basic definition of a chaotic system is a dynamic system with a specific set of properties:

  1. Sensitivity to initial conditions,
  2. Dense periodic orbits, and
  3. topological mixing

The property that we want to focus on right now is the
dense periodic orbits.

In a dynamical system, an orbit isn’t what we typically think of as orbits. If you look at all of the paths through the phase space of a system, you can divide it into partitions. If the system enters a state in any partition, then every state that it ever goes through will be part of the same partition. Each of those partitions is called an orbit. What makes this so different from our intuitive notion of orbits is that the intuitive orbit repeats. In a dynamical system, an orbit is just a set of points, paths through the phase space of the system. It may never do anything remotely close to repeating – but it’s an orbit. For example, if I describe a system which is the state of an object floating down a river, the path that it takes is an orbit. But it obviously can’t repeat – the object isn’t going to go back up to the beginning of the river.

An orbit that repeats is called a periodic orbit. So our intuitive notion of orbits is really about periodic orbits.

Periodic orbits are tightly connected to chaotic systems. In a chaotic system, one of the basic properties is a particular kind of unpredictability. Sensitivity to initial conditions is what most people think of – but the orbital property is actually more interesting.

A chaotic system has dense periodic orbits. Now, what does that mean? I explained it once before, but I managed to miss one of the most interesting bits of it.

The points of a chaotic system are dense around the periodic orbits. In mathematical terms, that means that every point in the attractor for the chaotic system is arbitrarily close to some point on a periodic orbit. Pick a point in the chaotic attractor, and pick a distance greater than zero. No matter how small that distance is, there’s a periodic orbit within that distance of the point in the attractor.

The last property of the chaotic system – the one which makes the dense periodic orbits so interesting – is topological mixing. I’m not going to go into detail about it here – that’s for the next post. But what happens when you combine topological mixing with the density around the periodic orbits is that you get an amazing kind of unpredictability.

You can find stable states of the system, where everything just cycles through an orbit. And you can find an instance of the system that appears to be in that stable state. But in fact, virtually all of the time, you’ll be wrong. The most minuscule deviation, any unmeasurably small difference between the theoretical stable state and the actual state of the system – and at some point, your behavior will diverge. You could stay close to the stable state for a very long time – and then, whammo! the system will do something that appears to be completely insane.

What the density around periodic orbits means is that even though most of the points in the phase space aren’t part of periodic orbits, you can’t possibly distinguish them from the ones that are. A point that appears to be stable probably isn’t. And the difference between real stability and apparent stability is unmeasurably, indistinguishably small. It’s not just the initial conditions of the system that are sensitive. The entire system is sensitive. Even if you managed to get it into a stable state, the slightest perturbation, the tiniest change, could cause a drastic change at some unpredictable time in the future.

This is the real butterfly effect. A butterfly flaps its wings – and the tiny movement of air caused by that pushes the weather system that tiny bit off of a stable orbit, and winds up causing the diversion that leads to a hurricane. The tiniest change at any time can completely blow up.

It also gives us a handle on another property of chaotic systems as models of real phenomena: we can’t reverse them. Knowing the measured state of a chaotic system, we cannot tell how it got there. Even if it appears to be in a stable state, if it’s part of a chaotic system, it could have just “swung in” the chaotic state from something very different. Or it could have been in what appeared to be a stable state for a long time, and then suddenly diverge. Density effectively means that we can’t distinguish the stable case from either of the two chaotic cases.

17 thoughts on “More about Dense Periodic Orbits

  1. jbw

    I think you are using the word “stable” incorrectly. All of the periodic orbits are unstable. Their instability is precisely your point, small perturbations cause trajectories to move away from the orbit. The property of orbits that you are calling “stable” is that they are invariant. If you start exactly on a periodic orbit, you remain on that orbit forever.

  2. Ivan

    The points of a chaotic system are dense around the periodic orbits.

    Sorry to nitpick, but I think it’s important to clarify that this is a colloquial usage of the word dense rather than a technical mathematical one. (I.e., “X is dense around Y” does not have an accepted mathematical meaning that I know of, although you seem to be using it as a synonymous phrase for the accepted wording “Y is dense in X”. It sounds like you’re saying that X is dense when you really mean that Y is dense.)
    However, your description which follows this statement is a perfectly good description of the fact that the periodic orbits are dense.

  3. --bill

    It’s not just that the periodic orbits are dense, it’s that there are periodic orbits of arbitrarily high period arbitrarily near any point. Or: given x and r and N, there exists y with |y-x|

  4. manuelg

    From the comments of earlier chaos post:–oh-my–oh-my#comment-2051899
    The textbook I used the first time I was learning about Chaos theory was “A First Course in Chaotic Dynamical Systems” by Robert Devaney (ISBN: 978-0201-554-069). It’s undergraduate level, but well written as I recall.
    It also keeps its focus on one dimensional discrete-time systems (i.e. iterating something that maps real numbers to real numbers), although it does cover a few other topics. The advantage of that is the systems can be analyzed explicitly without too much difficulty.
    Posted by: MPL | November 5, 2009 8:51 PM

  5. csrster

    I’m with jbw here. You seem to be using the word “stable” as synonymous with periodic, which is confusing.

  6. teramut

    Does that mean that for any periodic orbit, there is a point in that orbit where an arbitrarily small perturbation will derail the system? Can there be regions of the orbit where it is completely stable, or does the denseness also mean that every interval in the orbit has arbitrarily unstable points?

  7. mattheath

    Another nitpick: You seem to be using “partition” in a non-standard way. A partition of a set X is a collection of subsets of X such that each element of X is in exactly one of the subsets. So here the partition would be the collection of orbits, not an orbit.

  8. Ivan

    Oh, that’s good, Mark. Make another pointless post about the Cantor retards instead of replying to or correcting any of your earlier posts.

  9. Mark C. Chu-Carroll

    So what is it that you think I should be doing? What’s the glaring error that I haven’t corrected? What’s the urgent thing that I haven’t responded to?

  10. Ivan

    Well, you don’t seem interested in editing your post to correct any of the issues that have been pointed out in this thread, or the thing about two different notions of tree in the previous Cantor thread. I think these things are pretty important if you want mathematical newbies not to get confused unnecessarily– and I suppose I might be willing to call them glaring errors since, after all, a talented crank can crank out an entire system of crankery on the basis of smaller misunderstandings of technical terms.
    Regarding responding to things: see #1 and #8. Not exactly urgent, of course.
    I apologize for being rude in my previous comment.

  11. jbw

    So what is it that you think I should be doing?
    I am surprised you haven’t commented about your error in calling the orbits stable. You are a mathematician writing about mathematics and surely you know the importance of using the correct terms for well defined concepts.
    I really enjoy this blog. When I read about math that I am not familiar with I always assumed that you got the mathematics right, and when you made the inevitable mistakes you would correct them. If I read these comments without knowing this field, I wouldn’t know if the comments were nonsense or not.
    Since you asked, I think you should correct errors in the mathematics, so your readers can be assured they are not learning bad math from this blog.

  12. Anonymous

    I’m guessing the book’s title has some of these words in it: ‘dynamics’, ‘behavior’, ‘chaos’. Am I right?

  13. bware

    Actually, jbw and others, the author is using a correct version of the word stable (so are you).
    A stable orbit, in one sense, as you pointed out, refers to an orbit when perturbed slightly that trajectories move far away from that orbit.
    A slightly different sense would mean something like ‘an orbit around a stable equilibrium point’, i.e. an orbit that is itself a small enough deviation from a constant orbit. By small enough, we mean that we want the points of the orbit itself to remain in a bounded area. This is the only meaning you would meet in a simple classical mechanics class.
    In the first sense, we would instead want the points of the nearby orbits to remain close to points of the original orbit; however, neither the original or perturbed orbits would themselves have to be bounded. See ‘floating in a river’ example.
    With some mild conditions, the second sense of the word will require a periodic orbit. I’m not sure what the conditions are exactly. A counterexample would be the harmonic oscillator with friction, which is stable in both senses but not periodic (the motion slowly decays.)

  14. Muneeb Ahmed

    “every point in the attractor for the chaotic system is arbitrarily close to some point on a periodic orbit.”
    Does this mean that every point on the trajectory of the chaotic system through phase-space is in the neighborhood of a point on a periodic orbit?
    Ergo, that the trajectory of the attractor through phase space takes it through points (in the phase space) that may all be modeled as approximations of points on periodic orbits?
    I’m sorry if this is a noob question, I’ve only recently started studying this and have no source to crosscheck against.


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