{"id":779,"date":"2009-06-12T08:57:02","date_gmt":"2009-06-12T08:57:02","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2009\/06\/12\/defining-dynamical-systems\/"},"modified":"2009-06-12T08:57:02","modified_gmt":"2009-06-12T08:57:02","slug":"defining-dynamical-systems","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2009\/06\/12\/defining-dynamical-systems\/","title":{"rendered":"Defining Dynamical Systems"},"content":{"rendered":"

In my first chaos post<\/a>, I kept talking about dynamical systems<\/em> 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.<\/p>\n

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.<\/p>\n

The basic idea is pretty simple. A dynamical system is a system that changes
\nover time, and whose behavior can be (in theory) described a function that takes
\ntime as a parameter. So, for example, if you have a gravitational system which
\nhas three bodies interacting gravitationally, that’s a dynamical system. If you
\nknow 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.<\/p>\n

<\/p>\n

It’s important to understand, though, that as I mentioned in the first chaos
\npost: as is typical for mathematical things, most things are bad. Just because
\na function exists<\/em> doesn’t mean that it’s computable<\/em> or
\nderivable<\/em>. For most dynamical systems, we know that the system
\nis parametric in time, but we don’t know an equation for it.<\/p>\n

The most common case for interesting dynamical system that aren’t linear is
\nto describe the system in terms of differential equations. A differential
\nequation for a dynamical system basically says “Given the state of the system at
\ntime t, this equation tells you what the state of the system will be at time
\nt+ε”, where ε is an infinitesimally small period of time.<\/p>\n

To get a precise answer out of a differential equation, you need to be able
\nto integrate it. But most of the time, we don’t know how to integrate it
\nsymbolically. The closest we can come is to evaluate it as a series of
\nsteps, keeping the steps as small as possible. The result of doing this is not<\/em> exactly correct, but if you can get the time-steps short enough,
\nyou can get very close to the correct answer.<\/p>\n

For a lot of systems, this approach works really well. For one prominent
\nexample, it generally works quite well for N-body gravitational dynamics of
\nthings like the solar system. N-body systems are difficult and have some
\nseriously unstable points. But for many examples, with precise measurements and
\nsmall timesteps, you can get astonishingly accurate predictions using stepwise
\nevaluation of the differential equations. They’re very good, but far from
\nperfect. To give you a sense of what I mean by that: we can predict pretty much
\nexactly<\/em> where the earth will be at any point for the next 10,000 years.
\nBut there are several asteroids whose orbits come very close to earth (very
\nclose in astronomical terms that is), and we can’t be absolutely certain of
\nwhere they’ll be 30 years from now. The best we can do is talk in terms of
\nprobabilities.<\/p>\n

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.<\/p>\n

Using topology, you can describe almost anything continuous in terms of a
\nspace<\/em>. A topological space is a tricky concept, but the gist of it is
\nthat it’s an infinite set of objects (called points<\/em>), along with a
\nstructure that defines what objects are close to<\/em> one another. If you
\nwant more detail than that, then I’ve got a whole series of posts on topology
\nthat you can look at, starting here<\/a><\/p>\n

If you look at a complex system, you can define the set of states of that
\ncomplex system as the points of a space, and where points are close to each
\nother when there’s a short path through the states of the system from one
\nof those points to the other. If you define it so that it’s got the right properties, you end up with a topological space.<\/p>\n

To get from there to the phase space of a dynamical system, you need to add
\ntime<\/em> – the defining characteristic of a dynamical system is that it’s
\nparametric in time. That’s done by providing an evolution function<\/em>: a
\nmapping which, given any point p in the phase space of the dynamical system and
\nany interval of time, gives you another<\/em> point, p’ in the space. The meaning of the evolution function is that if you start the system in the
\nstate corresponding to the point p, and then you stop it after time t has passed, the state of the system will be p’.<\/p>\n

The evolution function is completely deterministic: given a precise point in
\nthe phase space, after a precise interval has passed, the system will
\nalways<\/em> wind up in a specific state. At this level of the system,
\nthere is nothing obviously chaotic, nothing uncertain, nothing random. The system is precise, fully defined, and fully deterministic.<\/p>\n

For many systems, the phase space is very clear and well defined, and
\nwe 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
\ndynamical systems is why chaotic systems were such a shock.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[77],"tags":[],"class_list":["post-779","post","type-post","status-publish","format-standard","hentry","category-chaos"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-cz","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/779","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=779"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/779\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=779"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=779"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}