The last major property of a chaotic system is topological mixing. You can
\nthink of mixing as being, in some sense, the opposite of the dense periodic
\norbits property. Intuitively, the dense orbits tell you that things that are
\narbitrarily close together for arbitrarily long periods of time can have
\nvastly different behaviors. Mixing means that things that are arbitrarily far
\napart will eventually wind up looking nearly the same – if only for a little
\nwhile.<\/p>\n
Let’s start with a formal definition.<\/p>\n
As you can guess from the name, topological mixing is a property defined Now, imagine that you’ve got a dynamical system whose phase space is The phase space is topologically mixing if, for any two open spaces A Now, let’s put that together with the other basic properties of All of this is a fancy and complicated way of saying that in a chaotic Anyway – I’ve spent more than enough time on the definition; I think I’ve So when you build a model of a chaotic system, you know that it’s going to From here, I’m going to move from defining things to analyzing things. Chaotic The last major property of a chaotic system is topological mixing. You can think of mixing as being, in some sense, the opposite of the dense periodic orbits property. Intuitively, the dense orbits tell you that things that are arbitrarily close together for arbitrarily long periods of time can have vastly different behaviors. Mixing means […]<\/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-843","post","type-post","status-publish","format-standard","hentry","category-chaos"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-dB","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/843","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=843"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/843\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=843"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=843"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=843"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nusing topology. In topology, we generally define things in terms of open sets<\/em>
\nand neighborhoods<\/em>. I don’t want to go too deep into detail – but an
\nopen set captures the notion of a collection of points with a well-defined boundary
\nthat is not<\/em> part of the set. So, for example, in a simple 2-dimensional
\neuclidean space, the contents of a circle are one kind of open set; the boundary is
\nthe circle itself. <\/p>\n
\ndefined as a topological space. The system is defined by a recurrence
\nrelation: sn+1<\/sub> = f(sn<\/sub>). Now, suppose that in this
\ndynamical system, we can expand the state function so that it works as a
\ncontinous map over sets. So if we have an open set of points A, then we can
\ntalk about the set of points that that open set will be mapped to by f. Speaking
\ninformally, we can say that if B=f(A), B is the space of points that could be mapped
\nto by points in A.<\/p>\n
\nand B, there is some<\/em> integer N such that fN<\/sup>(A) ∩ B &neq; 0. That is, no matter where you start,
\nno matter how far away you are from some other point, eventually<\/em>,
\nyou’ll wind up arbitrarily close to that other point. (Note: I originally left out the quantification of N.)<\/em><\/p>\n
\na chaotic system. In informal terms, what it means is:<\/p>\n\n
\ntrajectories are close together, at any time, they can<\/em>
\nsuddenly go in completely different directions.<\/li>\n
\ntheir trajectories stay far apart, eventually, they’ll
\nwind up in almost the same place.<\/li>\n<\/ol>\n
\nsystem, you never know what the heck is going to happen. No matter how long
\nthe system’s behavior appears to be perfectly stable and predictable, there’s
\nabsolutely no guarantee that the behavior is actually in a periodic orbit. It
\ncould, at any time, diverge into something totally unpredictable.<\/p>\n
\npretty well driven this into the ground. But I hope that in doing so, I’ve
\ngotten across the degree of unpredictability of a chaotic system. There’s a
\nreason that chaotic systems are considered to be a nightmare for numerical
\nanalysis of dynamical systems. It means that the most miniscule errors
\nin any aspect of anything will produce drastic divergence. <\/p>\n
\nbreak down. No matter how careful you are, even if you had impossibly perfect measurements,
\njust the nature of numerical computation – the limited precision and roundoff
\nerrors of numerical representations – mean that your model is going to break.<\/p>\n
\nsystems are a nightmare for modeling. But there are ways of recognizing when
\na systems behavior is going to become chaotic. What I’m going to do next is look
\nat how we can describe and analyze systems in order to recognize and predict
\nwhen they’ll become chaotic.<\/p>\n","protected":false},"excerpt":{"rendered":"