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.