topological mixing<\/li>\n<\/ol>\n The property that we want to focus on right now is the
\ndense periodic orbits.<\/p>\n
In a dynamical system, an orbit<\/em> 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<\/em>. What makes this so different from our intuitive notion of orbits is that the intuitive orbit repeats<\/em>. 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 beginning of the river.<\/p>\n An orbit that repeats is called a periodic orbit<\/em>. So our intuitive notion of orbits is really about periodic<\/em> orbits.<\/p>\n 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.<\/p>\n
A chaotic system has dense periodic orbits<\/em>. Now, what does that mean? I explained it once before, but I managed to miss one of the most interesting bits of it.<\/p>\n The points of a chaotic system are dense<\/em> around the periodic orbits. In mathematical terms, that means that every point in the attractor for the chaotic system is arbitrarily<\/em> 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.<\/p>\n 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 unpredictability.<\/p>\n
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<\/em> to be in that stable state. But in fact, virtually all<\/em> of the time, you’ll be wrong. The most minuscule deviation, any unmeasurably small difference between the theoretical stable state and the actual state of the system – and at some point, your 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.<\/p>\n What the density around periodic orbits means is that even though most<\/em> 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<\/em> isn’t. And the difference between real stability and apparent stability is unmeasurably, indistinguishably small. It’s not just the initial<\/em> 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 perturbation, the tiniest change, could cause a drastic change at some unpredictable time in the future.<\/p>\n 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.<\/p>\n
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<\/em> 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.<\/p>\n","protected":false},"excerpt":{"rendered":"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 […]<\/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-840","post","type-post","status-publish","format-standard","hentry","category-chaos"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-dy","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/840","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=840"}],"version-history":[{"count":2,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/840\/revisions"}],"predecessor-version":[{"id":3688,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/840\/revisions\/3688"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=840"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=840"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}