Talk:Orbit (dynamics)

Explain

Can we explain this better in English? -- The Anome

I think so - let me have a stab at it hawthorn

I'm not sure I understand how the given definition of closed orbits applies to continuous dynamical systems (ie, ODEs) ... the only orbits in such systems that contain finitely many points are stationary points ... right? It seems like closed orbits should be defined differently in the context of ODEs. Or am I missing something here? Maybe I'm not understanding the sense in which "infinite" is being used...? --Buggi22 05:47, 23 May 2005 (UTC)

The article could easilly be wrong at some level. I wrote it, and I'm not a mathematician, just a (now former) math student who took some dynamics. We have plenty of real mathematicians on Wikipedia, but none of them seem to have a strong interest in dynamics, so I tried to fill in gaps. Isomorphic 06:35, 25 May 2005 (UTC)

Comments on the rewrite

What is "discrete time"? The article probably had a slant toward discrete dynamics before, but now it sounds bizare. When you're iterating a map, there's no inherent assumption that your iterations have anything to do with time at all. That's just something that comes in when you're doing applications of dynamics. We need a more general definition. Isomorphic 19:30, 29 May 2005 (UTC)

Some authors tend to use orbit for maps and trajectories for flows, but it is common to speak of periodic orbits for flows. My intent was to include the continuous time definition of orbit in the article. The formal definition of an orbit uses the notion of a dynamical system, which is a manifold M together with a function f ^t(x) that describes how the points move around. The time parameter t ∈ T, which is one of four sets: the integers, the non-negative integers, the reals, or the non-negative reals. Sets with different additive structure have not as yet yielded any significant theory. I agree that the expression discrete time is a little strange, but it is also common usage in the dynamical systems literature.
  An orbit is a set parameterized by the t variable, time. An orbit or trajectory with initial condition u is the set

$\{x\in M|x=f^{s}(u)\,,s\in T\}$

Our challenge is to convert this mathematical definition into prose. XaosBits 13:43, 30 May 2005 (UTC)

Apologies. Now that I've gone back and reread bits of my old dynamics textbook, it's clear that your edits were fine and that I don't know what I'm talking about. No matter how much "discrete time" sounds to me like something from quantum mechanics, it is indeed the correct terminology. It's nice to finally have someone around Wikipedia who knows dynamics. Proceed, sir! I can worry about readability, if you worry about accuracy. I'm pretty good at explaining things in English once I understand them correctly myself. Isomorphic 04:05, 31 May 2005 (UTC)

I think your rewrite reads much better. XaosBits 01:14, 1 Jun 2005 (UTC)

Geometry

The new paragraph on geometry (or is it topology?) needs context. I understand the fact presented, but not why it's interesting or useful. Could you provide context or a better explanation? In the mean time I moved it to the bottom so as not to break the flow of the rest of the article. Isomorphic 05:26, 15 September 2005 (UTC)

It is a generalization of what is discussed in Polygon#Angles. I and others may think of expansions and more connections to other articles later.--Patrick 20:35, 15 September 2005 (UTC)

What I'm trying to say is, angles don't come to mind in the context of mathematical orbits. The concept of an orbit doesn't even require that you're in a metric space. It certainly doesn't guarantee Euclidean geometry. So talking about angles, and a property of those angles that doesn't even hold for most cases, seems beside the point. Isomorphic 01:52, 17 September 2005 (UTC)

I removed the paragraph on the turning angle. The turning angle (the winding number of the tangent vector) and other types of winding numbers seem appropriate topics in differential geometry, but are not directly applicable to trajectories of dynamical systems. (The article on winding numbers could use some work; it only takes the complex variables perspective.) There have been some attempts in the literature to use geometrical characterizations of the orbits of dynamical systems, but the results, while elegant, are specific to low-dimensional systems and do not generalize. The exception is the Maslov index of a closed curve.  XaosBits 02:52, 17 September 2005 (UTC)

Ok, I moved it to rigid body.--Patrick 14:47, 17 September 2005 (UTC)

period

I can't find the explicit definition of period here. --Adam majewski (talk) 19:14, 28 March 2012 (UTC)

Orbit cannot converge

Ii is written

It could be an asymptotically periodic orbit if it converges to a periodic orbit. ...

However, the orbit is defined to be the set (not sequence) of points of the trajectory; so it cannot converge (in the sense of limit, as linked) to anything. — MFH:Talk 00:51, 27 March 2022 (UTC)

Backward orbit typo (?)

When defining the orbit for a discrete time dynamical system, the backward subset is shown as follows: $\gamma _{x}^{-}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{f^{t}(x):t\geq 0\}$. Shouldn't it be $\gamma _{x}^{-}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{f^{t}(x):t\leq 0\}$? 2804:14C:BF27:4B2C:9A68:7D7A:F7F0:5521 (talk) 03:57, 11 February 2024 (UTC)