It’s been quite a while since my last chaos theory post. I’ve
been caught up in other things, and I’ve needed to do some studying. 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:
- Sensitivity to initial conditions,
- Dense periodic orbits, and
- 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 beginining of the river.
An orbit that repeats is called a periodic orbit. So
our intuitive notion of orbits is really about periodic
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
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 miniscule deviation, any unmeasurably small difference between
the theoretical stable state and the actual state of the system – and
at some point, you’re 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 pertubation,
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.