Talk:Dynamical system (definition)

Creation
I have created this article from the definitions that were added to the Dynamical system page. I modified the definitions of the smooth cases to use a manifold as the state space. I also moved Linas' measure theoretic definition to this page. &mdash; XaosBits 20:11, 15 October 2006 (UTC)

The claim
The claim that cellular automata are not dynamical systems because they are not invertible is a contentious one. Dynamical systems need not be invertible, although some of them are. See, e.g. "Introduction to the Modern Theory of Dynamical Systems" by Katok & Hasselblatt, Chapter 0, p2: "For a reversible system the transformations phi_t are defined for both positive and negative values of t and each phi_t is invertible. (My emphasis)" - i.e. dynamical systems, including cellular automata, need not be reversible. 139.184.30.17 16:48, 19 October 2006 (UTC)

A new section
I would like to add a section to show how a evolution function could be constructed, with particular emphasis on the physical origins of the concept and its relations to engineering applications. Is there any problem? Any criticism is welcome. :) Daniele Tampieri 11:45, 26 November 2006

Definition of I(x)
193.204.253.144 14:10, 16 January 2007 (UTC)The current definition of $$I(x)$$ is not very meaningful. $$I(x):= \{t \in T: (t,x) \in T \times M\}$$, as $$T \times M$$ is comprised of all the possible pairs in which the first element is in $$T$$ and the second in $$M$$, so basically couldn't we simply write that $$I(x):= T$$?


 * I agree with the previous comment: the section 'General definitions' is not clear. It would be useful for the reader to have a rough explanation, for instance that 'T' models the time, 'M' models space, and so on. At a first reading, I am not even sure what I(x) is: it is presented in such a way that we expect it to be used earlier on, which is not the case. — Preceding unsigned comment added by Nounec (talk • contribs) 14:52, 13 September 2011 (UTC)


 * The current definition $$\Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, $$ for $$\, t_1, t_2, t_1 + t_2 \in I(x)\,$$ is not good, because $$\Phi(t_1,x)=x_1 \in M$$, which is not necessary equal x. So $$t_2 \in I(x_1)$$, not $$I(x)$$. I think the correct definition must be:
 * $$\Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, $$ for $$\, t_1, t_1 + t_2 \in I(x)$$ and $$t_2 \in I(x_1)$$, where $$x_1 = \Phi(t_1,x)$$. — Preceding unsigned comment added by 91.122.87.233 (talk) 21:38, 14 September 2011 (UTC)

$$ T $$ doesn't have to be a monoid
The restriction that $$ T $$ must be a monoid strikes any kind of incomplete flow. For example, $$ dx/dt = x^2 $$ is (by this definition) not a dynamical system, since it blows up in finite time. The space $$ T $$ depends on each initial condition. SmaleDuffin 19:06, 20 April 2007 (UTC)

I changed the general definition to include incomplete flows. SmaleDuffin 17:44, 24 April 2007 (UTC

The term
The term "Dynamical systems" is not correct English. It seems to be a translation from another language. Traditionally the terms are "static" and "dynamic". "Dynamical" doesn't make any more sense than "statical".

A search of Amazon does show books with "dynamical systems" in the title -- about 900 of them. However, there are about 3,000 titles with "dynamic systems" in them. So that's about a 10 to 3 vote for "Dynamic systems" over "Dynamical".

Romeo and Juliette was not a "Romantical Tragedy", either. - EI 208.34.100.161 03:06, 2 November 2007 (UTC)


 * Dynamical system is the term as used by mathematicians; see the American Mathematical Society Mathematics Subject Classification: http://www.ams.org/msc/. 37-xx is "Dynamical Systems and Ergodic Theory."
 * SmaleDuffin 18:11, 2 November 2007 (UTC)


 * So, you would say "a classic mechanic system" and "mathematic proof"? You wouldn't. — Preceding unsigned comment added by 128.214.5.124 (talk) 23:45, 18 March 2012 (UTC)

As a researcher specializing in the field of Dynamical Systems, I confirm the above: The correct term is "Dynamical Systems", and it is used by all professionals I am aware of, including the British and American ones. — Preceding unsigned comment added by 128.214.129.220 (talk) 17:50, 3 February 2012 (UTC)


 * I can also confirm that 'dynamical' is now the correct usage, even though it seems ungrammatical to older ears. Just as computer scientists suddenly started referring to Graphic User Interfaces (GUIs) as 'Graphical' User Interfaces sometime in the 1980s, the phrase dynamic systems has now been replaced by dynamical systems. You can see the change in the titles of scientific books and papers if you look at their dates. Many other words have had superfluous 'ical' suffixes added in the last couple of decades. It sounds stupid to me too but English is a dynamical language and you have to roll with the changes if you want to survive. — Preceding unsigned comment added by 64.183.245.131 (talk) 18:48, 13 May 2013 (UTC)

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:48, 10 November 2007 (UTC)

Definition of Time and Example
The article currently defines time as a monoid, and the "reached time" for a given initial state as subset of the monoid. This very general definition seems to insist on the fact that time can be something else than $$\mathbb{N}$$ or $$\mathbb{R}$$. I wonder if some example can be added to illustrate this interesting claim. In particular, are there examples with where time is a:  non-commutative monoid ? non-Archimedian monoid ? non-totally-ordered monoid ? non-ordered monoid ? etc. Depending on this, it may be possible to say more about "reached times", whether they look like intervals or anything with more structure than a simple subset. Frozsyn (talk) 13:36, 10 June 2010 (UTC)

In philosophy
There exists a philosophical definition which is more general and much more clear than the other technical formalisms. The suggested definition is the following:
 * A system is a dynamical object which can be defined as an entity, composed of a compplex structure performing a process, which is controlled, and operating within an environment. (source: Frederick Betz,, 2010)

Possibly, it would be better to add it to the Wp article titled systems theory or dynamic system, but it can be taken into account that Maths and Philosophy are historically more linked each other than the other sciences.Philosopher81sp (talk) 23:38, 18 January 2021 (UTC)