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Rewrite of 11 June 05
Dynamical systems is a vast subject and I could not finish this article. I ran out of steam. I have tried to write the article using a minimal amount of formalism. My hope is that the more technical stuff will be in the linked articles. Minimizing formalism turned out to be harder than I expected. For example, it took me close to an hour to write the definition in the first sentence:
- A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space.
There are many missing parts to the article, most notably a discussion of hyperbolic systems, and of stable and unstable manifolds and their intersections. There is no history of the subject. The article also needs some illustrations. XaosBits 14:05, 11 Jun 2005 (UTC)
- I am very impressed with your rewrite; well done! However, I don't understand the second sentence of the following fragment:
- "Linear systems are simpler than more general dynamical systems. But it is not the linearity that makes them simple, it is the structure of the orbits in phase space. There are coordinate transformations that would not change the structure of phase space, but could make it into a nonlinear system."
- I thought that linear systems are so simple because one can solve them explicitly. They are not very simple; for instance, you can have a centre which is a rather non-generic structure. I am certain you have a good reason for what you wrote down, but I am just curious what it is, so could you please enlighten me (in technical language, if necessary)?
- A small remark: my browser doesn't show ₀ (₀). What is this supposed to be, and is it necessary to use it, as other people's browsers may have the same problem. I also notice that you did not include any examples. Was this a conscious decision or are they to be included in the illustrations you mention above?
- Thanks again, and I look forward to reading more of your work. -- Jitse Niesen 08:20, 12 Jun 2005 (UTC)
- Now that I re-read that sentence, I agree, its wrong (I'm redefining linear). I deleted it. What I had in mind when I was writing that was to point out that it is the structure of the vector field that makes analysis difficult. So if I pick a coordinate change that is nonlinear and apply it to a x-dot = v(x) system, I will end up with a non-linear system that is no more interesting than the linear one.
- The lack of examples is lack of energy on my part. There should be examples and illustrations.
- The ₀ is a zero subscript. I've replaced them in the main article.
- XaosBits 16:10, 12 Jun 2005 (UTC)
On the first version
Great that somebody started this entry. Should we have the article under dynamic system or dynamical system.
Google gives 35400 hits for dynamical system and 59200 for dynamic system. Are this terms real synonyms or is there a difference in meaning?
--- Good question, i would say there are synonyms, but since iam not a native english speaker iam not sure. Any other opinions?
--Thnord 22:34, 25 Apr 2004 (UTC) "Dynamical system" One of the foremost researchers, Field medal winner Stephen Smale used that term. That's good enough reason for me.
- The Mathematics Subject Classification, used in Mathematical reviews, also uses Dynamical system (category 37). -- Jitse Niesen 11:05, 26 Apr 2004 (UTC)
I removed the phrase 'only involves the variable's current value' from the intro, as the rules defining a dynamical system invariably involve constants too. I changed it to 'is defined in terms of the variable's current value'. Chopchopwhitey 00:36, 13 Mar 2004 (UTC)
discrete dynamical systems?
I changed the definition back. A dynamical system is a smooth function from a manifold to itself dependent on one parameter, time. While the manifold could be replaced by a more general set, loosing smoothness takes one to ergodic theory or symbolic dynamics. The parameter needs to be a monoid traditionally written in additive form. One could just formalize that definition, but the challenge is to re-write it so that someone with high school mathematics can follow the definition.
From what I replaced, I liked the use of relationship instead of function and responding to their own values, which emphasizes the mapping a point of the manifold back onto itself. But I could not get it to work. Maybe someone else could try. XaosBits (talk) 05:53, 29 December 2015 (UTC)
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