Another one of the fundamental properties of a chaotic system is

*dense periodic orbits*. It's a bit of an odd one: a chaotic

system doesn't have to have periodic orbits *at all*. But if it

does, then they have to be dense.

The dense periodic orbit rule is, in many ways, very similar to the

sensitivity to initial conditions. But personally, I find it rather more

interesting a way of describing key concept. The idea is, when you've got a

dense periodic orbit, it's an odd thing. It's a repeating system, which will

cycle through the same behavior, over and over again. But when you look at a

state of the system, you can't tell which fixed path it's on. In fact,

miniscule differences in the position, differences so small that you can't

measure them, can put you onto dramatically different paths. There's

the similarity with the initial conditions rule: you've got the same

basic idea of tiny changes producing dramatic results.

In order to understand this, we need to step back, and look at the some

basics: what's an orbit? What's a periodic orbit? And what are dense

orbits?

To begin with, what's an orbit?

If you've got a dynamical system, you can usually identify certain

patterns in it. In fact, you can (at least in theory) take its

phase space and partition it into a collection of sub-spaces which

have the property that if at any point in time, the system is in a

state in one partition, it will *never* enter a state in any

other partition. Those partitions are called *orbits*.

Looking at that naively, with the background that most of us have

associated with the word "orbit", you're probably thinking of orbits as being

something very much like planetary orbits. And that's not entirely a bad

connection to make: planetary orbits *are* orbits in the

dynamical system sense. But an orbit in a dynamical system is more like the

*real* orbits that the planets follow than like the idealized ellipses

that we usually think of. Planets don't really travel around the sun in smooth

elliptical paths - they wobble. They're pulled a little bit this way, a little

bit that way by their own moons, and by other bodies also orbiting the

sun. In a complex gravitational system like the solar system, the orbits

are complex paths. They might never repeat - but they're still orbits: a state

where where Jupiter was orbiting 25% closer to the sun that it is now

would never be on an orbital path that intersects with the current state of

the solar system. he intuitive notion of "orbit" is closer to what

dynamical systems call a *periodic* orbit: that is, an orbit that

repeats its path.

A *periodic orbit* is an orbit that repeats over time. That is,

if the system is described as a function f(t), then a periodic orbit is

a set of points Q where ∃Δt : ∀q∈Q: if f(t)=q,

then f(t+Δt)=q.

Lots of non-chaotic things have periodic orbits. A really simple

dynamical system with a periodic orbit is a pendulum. It's got a period,

and it loops round and round through a fixed cycle of states from its

phase space. You can see it as something very much like a planetary orbit,

as shown in the figure to the right.

On the other hand, in general, the real orbits of the planets in the solar

system are *not* periodic. The solar system never passes through

*exactly* the same state twice. There's no point in time at which

everything will be exactly the same.

But the solar system (and, I think, most chaotic systems) are, if not

periodic, then *nearly* periodic. The exact same state will never occur

twice - but it will come arbitrarily close. You have a system of orbits that

look almost periodic.

But then you get to the *density* issues. A dynamical

system with *dense* orbits is one where you have lots of different

orbits which are all closely tangled up. Making even the tiniest change

in the state of the system will shift the system into an entirely different orbit,

one which may be dramatically different.

Again, think of a pendulum. In a typical pendulum, if you give the pendulum

a little nudge, you've changed its swing: you either increased or decreased the amplitude

of its swing. If it were an ideal pendulum, your tiny nudge will *permanently*

change the orbit. Even the tiniest pertubation of it will create a permanently

change. But it's not a particularly *dramatic* change.

On the other hand, think of a system of planetary orbits. Give one of the planets

a nudge. It might do almost nothing. Or it might result in a total breakdown

of the stability of the system. There's a very small difference between a path

where a satellite is captured into gravitational orbit around a large body, and

a path where the satellite is ejected in a slingshot.

Or for another example, think of a *damped driven pendulum*. That's one

of the classic examples of a chaotic system. It's a pendulum that has some force that acts to reduce the swing when it gets too high; and it's got another force that ensures that it keeps swinging. Under the right conditions, you can get very unpredictable behavior. The damped driven pendulum produces a set of orbits that really demonstrate this, as shown to the right. Tiny changes in the state of the pendulum put you in different parts of the phase space very quickly.

In terms of Chaos, you can think of the orbits in terms of an attractor.

Remember, an attractor is a black hole in the phase space of a system, which

is surrounded by a basin. Within the basin, you're basically trapped in a

system of periodic orbits. You'll circle around the attractor forever, unable

to escape, inevitably trapped in a system of periodic or *nearly* orbits.

But even the tiniest change can push you into an entirely different

orbit, because the orbits are densely tangled up around the attractor.

Can a ball sitting motionless on a plane be considered a degenerate case for a dense periodic orbit? If I move it a little bit, it can never get back to a different point because its motionless.

Would every point in the phase space be considered an attractor?

Also, the image for the pendulum is not showing up.

I think you glossed over the dense

periodicorbits bit too much. More formally, it's that the set of points with periodic orbits are dense. That is, given any point in the phase space, there is a point with a periodic orbit within any given distance (no matter how small).That is, even though most points do

nothave periodic orbits, there are always periodic points very nearby.@2:

Thank you for that!

I'm learning chaos theory as I'm writing these posts. And frankly, none of the sources that I'm reading said that in such a clear way. I was misunderstanding it - but I can see how the formal prose really does end up meaning that. I'll post an update later, with a thank you to you.

You'll probably want to convert that TIFF to a PNG. Firefox won't display it, so I'm guessing most browsers won't either.

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.

So this all reminds me of my first (unwitting) exposure to chaotic systems, one that I failed to realize the full implications of until far later.

In early high school, for fun I wrote a program that (I thought) simulated a whole bunch of planets that all exercised gravitational attraction to each other. In essence, I was trying to solve the many-bodies problem, but without knowing what it was called or that it was more or less impossible. heh...

Being the naive young student I was, I simply looped through each body, calculated the distances to all the other masses, multiplied the mass by a constant and divided by the square of the distance, and updated the velocities accordingly. I didn't even think about sampling frequency at first...

The result was something that looked cool most of the time, but whenever any of the two masses got close together it would do really screwy stuff. Not having

anyknowledge of calculus at the time, except that it was a word and it existed, it took a bit of thinking on my part to puzzle out that, in the interval between updates, the positions would be changing, and hence the forces changing dramatically as well.Since I only wanted something that looked cool, I just reduced the sampling interval until I didn't tend to get weird discontinuities. But I remember thinking to myself, "Man, how the hell do they solve this problem?! You'd have to keep reducing the sampling interval until it was infinitely small!"

It wasn't until many years later that I learned about calculus, and later of the many-bodies problem. Heh.. shouldn't have been so ambitious.

just to clarify, you stated that "the planetary orbits are not periodic.....etc".

Unfortunately, you didn't mention with respect to which particular coordinate system they aren't. If you are saying "the planets' orbits ane 'only nearly' periodic, since they don't pass through the EXACT same spatial point after one period of time", then I guess I will point out that your spatial coordinates are dependant upon your temporal coordinates.

Since every (spatial) point in the WHOLE UNIVERSE is accelerating away from every other point at an ever increasing rate (with respect to galaxies and what not...), one can't say the orbits are even slightly (spatially) periodic--think of riding on a merry go round that is not rolling on the ground: you keep going around in circles, but to someone watching you from a point not on the merry-go-round, you are going to be moving up and down regularly, but also forward along the horizontal axis. if you were to trace your path (in a constant linear velocity situation) you would see what's called "cycloidal motion" kind of like what a bicycle's reflectors do when your headlights shine on a bike rider at night: a linearly travelling loop-de-loop motion. The only periodicity in this situation is in the vertical axis, where the observed point can only oscillate between a maximum height of the top of the tire and minimum height of the bottom of it. With respect to that coordinate system, the vertical periodicity IS EXACT. the tire cannot grow or shrink (beyond molecular shrinkage or expansion due to the effect of the road temperature and friction between the tire and the road creating tiny differential changes in the instantaneous pressure of the the tire's tube.......)

If you really wanna look at the dynamics of the solar system that way, there cannot be ANY periodicity whatsoever, since (general) relativity says that in reference frames accelerating WITH RESPECT TO ANY OTHER FRAME, we cannot apply the same laws of physics that we normally do. Since the magnitude of the acceleration of the universe is not increasing in a constant fashion(it's getting faster and faster), that means that we arent even able to pick any reference point with which to start referencing from!

In fact, nothing in the entire universe has ever even crossed the same point in space more than once! The whole point of the mathematics behind the sort of systems and situations is so that we can predict things in our own, non-inertial, reference frame, and use those predictions in our everyday lives.

Mathematicians need to be less nit picky. You don't need perfect grammar to communicate in a language.

@7,

The objections you raise are interesting, but not what is being discussed here. In fact, even in a completely deterministic Newtonian model of orbital dynamics, with no Relativity effects at all, and some fixed reference frame, orbits in systems with more than three bodies

stillaren't typically periodic.This is a math blog, not a language blog. Mathematicians aren't "nit picky" for the hell of it---there's a reason.

@brian

Center-of-mass coordinates for the solar system would be the natural implication. In a thought experiment we can ignore the rest of the universe (and to good approximation we can do that anyway).

There isn't really such a thing as a "point in space." There are only points in spacetime. You may however be interested in comoving coordinates. Unfortunately, if your assertion is rephrased in comoving terms, it's false.

Ourlocal galactic cluster is moving at a pretty good clip with respect to the local comoving frame, though.I would not use gravitational systems as an example of chaotic behavior, myself. Sure, small differences in initial conditions will result in large differences in outcomes if you shoot a satellite past a planet, but this is not chaotic behavior. (You could not actually put it into orbit using gravity alone, no matter what path you chose.) The size of the differences in outcomes is not what makes a system chaotic. What is important is that if you vary initial conditions slightly, the outcome might change dramatically, but it will change in a predictable way. If the system were chaotic, you would not be able to do this.

If, say, an asteroid were to collide with the moon, it would certainly change the precise location of the moon at some future time, but not a whole lot, and not in a non-linear fashion. In a chaotic solar system, it might end up in orbit around Jupiter.

I recently wrote an article for the Texas Advanced Computing Center about methods developed by Bruce Boghosian at Tufts to discover unstable periodic orbitals in turbulent systems.

http://www.tacc.utexas.edu/news/feature-stories/2010/the-skeleton-of-chaos/

Let me know what you think.