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
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,
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 periodic orbits 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 not have periodic orbits, there are always periodic points very nearby.
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 any knowledge 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.
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 still aren't typically periodic.
Mathematicians need to be less nit picky. You don't need perfect grammar to communicate in a language.
This is a math blog, not a language blog. Mathematicians aren't "nit picky" for the hell of it---there's a reason.
you didn't mention with respect to which particular coordinate system
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).
nothing in the entire universe has ever even crossed the same point in space more than once
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. Our local 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.
Let me know what you think.