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.
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How much of the uncertainty of the position of the asteroid is due to the failure of the discrete integration and how much is due to the uncertainty of the "real" environment? You seem to be saying it is a failure of the *math*.
After all, the asteroid may be influenced by a rather small body unknown to us, or some affect like the Pioneer Anamoly may mean that something more than gravity is affecting things.
I'm excited for when you get formal! I only recently discovered your blog, and I love it. It's nice to see people being somewhat comprehensive with regard to the mathematics of everyday life.
I'm a D.Sc. Systems Engineer now working in Health Care. Because I studied this all from an Engineering perspective, rather than a Mathematics perspective (or I should say, an Analytical perspective), I rarely dealt with Chaos. We just don't need to worry about it except in vary rare cases. For the most part, if we can't find a closed form solution, we'll do what you say: step-wise repeated numerical integrations.
Luckily, for many real world solutions in the engineering field, closed form solutions exist and are easy. Thermostats, cruise control, auto-pilots, all these are dynamical systems with closed form solutions we can manage in phase space.
OH, and do be careful with the word 'complex'! When doing frequency domain stuff, which is a phase space, we are working in a complex domain, which is REALLY not the same thing as a 'complicated' domain.
OH, and do be careful with the word 'complex'!
Unfortunately, we are stuck with that term. People use it in the layman's sense to describe systems with lots of complicated interactions, whether or not the system involves variables which may be complex in a strictly mathematical sense. If anything, I would argue that "complex number" (a term we use for historical reasons) is something of a misnomer, because they are much less complicated than a lot of other things mathematicians, let alone other scientists and engineers, have to deal with.
Yes, AnyEdge, after all that caused the Eastern European Airliner to crash. All the Poles were on the right hand side of the S-Plane.
Yep, KeithB, misplaced poles can suck. Why just recently I had a tent collapse....
I agree with you regarding the appropriateness of the language. After all, many type of numbers are far more complicated than the ones we find on say, Z[i]. And I float against the current as an engineer, because I will not use 'j' for the imaginary number (unless I'm fiddling around with i,j, and k.). But that's the way it formed, and this post regardsa situation where the imprecision could matter.
wow I suck at humour.
The way I heard it was something like:
Why did the flight from Warsaw to Paris crash after the pilot announced you could see the Eiffel Tower out the right side windows?
-- too many Poles in the right half plane
We are chaotic beings of chaotic cells thinking with chaotic brains, who evolved in a chaotic ecosystem, interacting in a chaotic society, embedded in a chaotic cosmos!
My 2nd cousin and frequent co-author, tenured Professor Dr. Philip Vos Fellman (Southern New Hampshire University), has been telling his students and friends for 2 decades that5 the 3 greatest discoveries of the 20th century were Relativity, Quantum Mechanics, and Chaos. One some days, I agree with that list of "top 3." Our Global Recession was created by people who didn't know about Chaos. We pay the price.
My PhD research (1973-1977) was about living metabolic systems as dynamical systems operating "At the Edge of Chaos" -- but the vocabulary did not then exist.
"I'm doing an on-line course for a school in Maryland and they let me set up a blog site (Chaos, Complexity, Modeling, Simulation and Strategic Management) at http://leadershipinglobalenterprise.ning.com/profiles/blogs/chaos-compl…
They are using a slightly different technology than that which I am familiar (ning) and it seems to require that you set up a login and password, probably to get you onto their platform."
I have always had trouble wrapping my brain around the notion that all of the universe started as a small point of energy. For one thing, our measurement of distance is based on the rate of light propagation, so how can you know the size of the universe at such small sizes? This got me to hypothesize that the universe is not expanding, but that its "spaces" are dividing into ever smaller proportions of the universe. Since we are in the universe, we perceive the division as expansion.
I somewhat agree, but it still seems that even if the universe was not infinitely small, it still appears to have been infinitely dense which is still a difficult to handle singularity.
It's been known for a long time that your step sizes when you use numerical methods can induce chaos in your solutions, but also there's the idea that with inherent complexity and discretization complexity - I think (probably wrong, but you never know) that Corless and Essex (?) about 20 years ago talked about the idea that you could also suppress chaos in your numerical solutions.
What this means for your simulations of a (potentially) chaotic system is that you may have chaos or you may not depending on your simulation and also you may or may not have inherent chaos built into the system itself. In other words, unless you analyze *very* carefully, you could have both of those situations you describe - uncertainty due to the dynamical system or uncertainty due to the numerical integration. Cool eh? :-)
Thanks for the great posts...
Re #1, #13:
I believe that the uncertainty in Mark's asteroid example is not due to chaos or computational error but rather to observational/experimental uncertainty. Whatever measurements you take to measure the state of an asteroid are only going to be so accurate, as a purely practical, not theoretical matter. If your theory, combined with computation says its a close call, but the input data to that computation (your actual measurements) were only so-so accurate, then you can't really be sure what's going to happen.
At least that's how I read it.
Actually, the uncertainty in the asteroid example is due to all of those factors: measurement error, computational error, and chaos.
When it comes to orbital mechanics, small pertubations can have dramatic results. If it weren't for chaotic factors, the measurements we can take of asteroids would be good enough for ruling out most encounters with asteroids. But the orbits of the asteroids are constantly perturbed in small ways by encounters with other asteroids, gravitational interactions with the planets, etc. Those pertubations, even though they're extremely small, can be enough to create a significant change. The gravitational change from a close pass with another asteroid can produce a change in the orbital radius of a couple of meters; that couple of meters can easily grow over the course of a decade or two to 50,000 kilometers - which is well within the range of uncertainty of our measurements.
A very nice visual introduction to dynamics and chaos - but without all of "scary" math - can be found in the series of books by Abraham and Shaw entitled "Dynamics - The Geometry of Behavior". Worth at least a look by anyone interested in the topic.
I think the typical example for the confluence algorithms, step sizes, and chaos is the set of "root finding" algorithms. With variation in step size, initial conditions, simple algorithms like Newton's method can start returning chaotic output, instead of converging to one of the roots.
A quick search of google for Newton's method + chaos, shows lots of links if you're interested.
I think you are right that a lot of the uncertainty of asteroid trajectories are due to unknown objects acting upon them AND the standard deviation inherent in calculating conversions.