In my first chaos post, I kept talking about dynamical systems without bothering to define them. Most people who read this blog probably have at least an informal idea of what a dynamical system is. But today I’m going to do a quick walkthrough of what a dynamical system is, and what the basic relation of dynamical systems is to chaos theory.
The formal definitions of dynamical systems are dependent on the notion of phase space. But before going all formal, we can walk through the basic concept informally.
The basic idea is pretty simple. A dynamical system is a system that changes
over time, and whose behavior can be (in theory) described a function that takes
time as a parameter. So, for example, if you have a gravitational system which
has three bodies interacting gravitationally, that’s a dynamical system. If you
know the initial masses, positions, and velocities of the planets, the positions of all three bodies at any future point in time is a function of the time.
It’s important to understand, though, that as I mentioned in the first chaos
post: as is typical for mathematical things, most things are bad. Just because
a function exists doesn’t mean that it’s computable or
derivable. For most dynamical systems, we know that the system
is parametric in time, but we don’t know an equation for it.
The most common case for interesting dynamical system that aren’t linear is
to describe the system in terms of differential equations. A differential
equation for a dynamical system basically says “Given the state of the system at
time t, this equation tells you what the state of the system will be at time
t+ε”, where ε is an infinitesimally small period of time.
To get a precise answer out of a differential equation, you need to be able
to integrate it. But most of the time, we don’t know how to integrate it
symbolically. The closest we can come is to evaluate it as a series of
steps, keeping the steps as small as possible. The result of doing this is not exactly correct, but if you can get the time-steps short enough,
you can get very close to the correct answer.
For a lot of systems, this approach works really well. For one prominent
example, it generally works quite well for N-body gravitational dynamics of
things like the solar system. N-body systems are difficult and have some
seriously unstable points. But for many examples, with precise measurements and
small timesteps, you can get astonishingly accurate predictions using stepwise
evaluation of the differential equations. They’re very good, but far from
perfect. To give you a sense of what I mean by that: we can predict pretty much
exactly where the earth will be at any point for the next 10,000 years.
But there are several asteroids whose orbits come very close to earth (very
close in astronomical terms that is), and we can’t be absolutely certain of
where they’ll be 30 years from now. The best we can do is talk in terms of
To reiterate: a dynamical system is basically a system that’s parametric in time. But for chaos theory, we want to describe it in terms of a phase space. To get to the phase space, you need to think of it in terms of topology.
Using topology, you can describe almost anything continuous in terms of a
space. A topological space is a tricky concept, but the gist of it is
that it’s an infinite set of objects (called points), along with a
structure that defines what objects are close to one another. If you
want more detail than that, then I’ve got a whole series of posts on topology
that you can look at, starting href="http://scienceblogs.com/goodmath/2006/08/topological_spaces.php">here
If you look at a complex system, you can define the set of states of that
complex system as the points of a space, and where points are close to each
other when there’s a short path through the states of the system from one
of those points to the other. If you define it so that it’s got the right properties, you end up with a topological space.
To get from there to the phase space of a dynamical system, you need to add
time – the defining characteristic of a dynamical system is that it’s
parametric in time. That’s done by providing an evolution function: a
mapping which, given any point p in the phase space of the dynamical system and
any interval of time, gives you another point, p’ in the space. The meaning of the evolution function is that if you start the system in the
state corresponding to the point p, and then you stop it after time t has passed, the state of the system will be p’.
The evolution function is completely deterministic: given a precise point in
the phase space, after a precise interval has passed, the system will
always wind up in a specific state. At this level of the system,
there is nothing obviously chaotic, nothing uncertain, nothing random. The system is precise, fully defined, and fully deterministic.
For many systems, the phase space is very clear and well defined, and
we can perform computations in it with great precision. Just for example, there are lots of linear dynamical systems, and they’re perfectly stable. In fact, you can make the argument that the ease with which we can analyze linear
dynamical systems is why chaotic systems were such a shock.