Talk:Dynamical system

From Wikipedia, the free encyclopedia
Jump to: navigation, search
          This article is of interest to the following WikiProjects:
WikiProject Mathematics (Rated C-class, Top-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
C Class
Top Importance
 Field: Mathematical physics
One of the 500 most frequently viewed mathematics articles.
This article has comments.
WikiProject Physics (Rated C-class, Mid-importance)
WikiProject icon This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
C-Class article C  This article has been rated as C-Class on the project's quality scale.
 Mid  This article has been rated as Mid-importance on the project's importance scale.

This article has comments here.

WikiProject Systems (Rated C-class, Top-importance)
WikiProject icon This article is within the scope of WikiProject Systems, which collaborates on articles related to systems and systems science.
C-Class article C  This article has been rated as C-Class on the project's quality scale.
 Top  This article has been rated as Top-importance on the project's importance scale.
Taskforce icon
This article is within the field of Dynamical systems.

This article has comments here.

This article has an assessment summary page.

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?

The term discrete dynamical system redirects here, but there is (almost) no discussion of the topic here. Would be nice for somebody to write it. Dmharvey Talk 5 July 2005 19:43 (UTC)

Formal definition

Is there some reference for this formal definition? It seems to be very similar to the definition for measure-preserving dynamical systems, and I seem to remember that not all systems are measure-preserving (take for instance the system \dot{x} = 1 with x(t) in R; what is the measure?). I have no books at hand though, so I can't check it myself at the moment. -- Jitse Niesen (talk) 12:52, 26 August 2005 (UTC)

As long as you are not at a singular point, one can create a measure that is preserved. If f^t is the evolution map, then a density a(x) gets mapped to
 \int dy \, a(y) \delta\left( x - f^t(y) \right) \,.
A pre-factor |f'(y)| can be added in the integral so that density a is invariant.
A formal definition for a dynamical system may be motivated by ergodic theory or by differential equations. When defining a dynamical system from the perspective of differential equations, one also has to decide if evolution functions arising from partial differential equations are going to be included or not. The choice depends on what part of the theory one wants to develop. The richest theory requires a dynamical system to be the tuple  \langle \mathcal{M}, f , \mathcal{T}\rangle , with \mathcal{M} a manifold (locally a Banach space or Euclidean space), \mathcal{T} the domain for time (non-negative reals, the integers, ...) and f^t a diffeomorphism of the manifold to itself. With the differentiability results about conjugations, fibrations, and invariant measures can be developed. In Smale's Bulletin article he speculated that parts of the theory could be developed when \mathcal{T} is a non-commutative group. I never saw anyone follow up on this suggestion.
The usual definitions exclude cellular automata (CA). CA seem to have many of the properties of dynamical systems, but the resemblance has not been extended to the level of a rich collection of theorems that apply both to CA and differential equations.    XaosBits 22:15, 27 August 2005 (UTC)
I found the article measure-preserving dynamical system only after adding the section on the "formal definition"; otherwise I wouldn't have added it. I've never seen a non-measure preserving system. Additional generalization would be great, but it would then be "original research" on my part, and my original research isn't very good. I don't have a reference; the only reference I have mentions it "in passing", and is unsuitable for inclusion here. linas 00:51, 30 August 2005 (UTC)
As if by magic, a reference appears before me ... added to measure-preserving dynamical system. This reference only makes the narrow definition, without claiming it's the "theory of everything" the way that the paragraph "formal definition" implies. linas 04:20, 30 August 2005 (UTC)
I would like to suggest a re-write of the formal definition section. There are really two definitions in use for dynamical systems, but it may be too subtle of a point to make in the Wikipedia. Maybe the point should not even be raised in the article. I would like to hear your comments. XaosBits 11:27, 31 August 2005 (UTC)
I'm feeling sleepy right now ... but after a quick skim, that looks reasonable. The distinction is not too fine a point; the only remark being, if this article gets too long, then it should be split up (with finer points in some advanced article). (WP has 10K math articles; fine points are encouraged). Besides, the second half of what I wrote about continuous systems was a hack job of handwaving, so yes, do cut it out. linas 05:45, 1 September 2005 (UTC)

"Dynamical system" definition excludes e.g. cellular automata — is this correct?

The definition given gives the impression that a dynamical system must be continuous in both space continuous time. Why this restriction? It was my impression that, for example, cellular automata (in which space and time are discrete) could be with perfect validity described as dynamical systems. WebDrake 15:25, 12 October 2005 (UTC)

No. discrete systems, e.g. symbolic dynamics, are arguably a part of the field of dynamical systems. The reason that cellular automata are usually excluded is because they maintain internal state, and not because they are discrete. In particular, both discrete and continuous dynamical systems can be given topologies, and topology is kind of a very fundamental part of how these systems can be understood. I am not aware of how cellular automata can be topologized. (If they can be, let me know how). linas 00:47, 13 October 2005 (UTC)
Another reason to exclude cellular automata is that most of the theory for dynamical systems does not extend to cellular automata. For example, there are no tangent spaces, nor stable and unstable manifolds. There is little geometry and the topology, as pointed out above, is not interesting. The structure of the phase space of cellular automata is limited: the phase space is a graph with possibly many connected components and each component is a cycle with trees attached to it.
Note that the definition allows time to be discrete. XaosBits 01:41, 13 October 2005 (UTC)
OK, I see what you're getting at here. However, it seems to me that this distinction is a pure-mathematical one; my impression is that elsewhere (e.g. in physics) the term is used somewhat more loosely. Maybe worth discussing?
Linas, just to make sure I understand, can you clarify the point about CA and "maintaining internal state"? I guess you are talking about update rules since I don't see how the space in which CA are defined (a list of locations i each associated with a value from some symbol set) is in itself distinct from the dynamical systems definition given here.
Just to note, my personal particular interest here is that one of the subjects I write on is self-organized criticality and I'm wanting to resolve the distinction between my casual use of "dynamical system" in the article and the stricter definition I find here. Any advice? — WebDrake 11:23, 14 October 2005 (UTC)
Two remarks: first, what XaosBits and I are saying are that CA are not normally considered to be dynamical systems, although I suppose some handwaving and blustering would be allowed.
Second remark: By "internal state", I mean what is usually meant: the state of the machine. A standard definition of a discrete-time dynamical system is given in measure-preserving dynamical system. You'll notice that the necessary ingredients are a topology given by a sigma-algebra, a measure, and a shift operator (aka time evolution operator). I don't know how to convert a generic CA into a sigma-algebra, nor do I know how to add a measure onto that topology. This could, of course, be my ignorance. I do know that tiling spaces can be constructed out of L-systems, (for example, the penrose tiling is an L-system, among other things). And I know that L-systems are described in terms of a grammer, and so I suppose that maybe if you had some general CA with some state transition rules that could be written as a grammer, then you could define a corresponding L-system, argue that the L-system is a tiling space, and thus has the usual metric on a tiling space, and thus we now have a topology, a measure, and a shift operator, and thus, the damn thing is a dynamical system. But these last sentences are pure original research on my part; I have no idea if this can actually be made to work; or what rules may need to be bent to make it work; I certainly have never read anything that connected all these dots. (See for example, subshift of finite type for some of these steps.) linas 23:47, 14 October 2005 (UTC)
p.s. I had a chance to ask Per Bak to be my thesis advisor in the early 80's, but muffed my chance; back then, chaos was considered to be a career-damaging research topic. Oh well. linas 23:53, 14 October 2005 (UTC)
p.p.s. I'm currently attending some lectures on the geometry (cohomology) of tiling spaces, focusing on those generated by substitution grammars, so I'll pop the question there. linas 00:01, 15 October 2005 (UTC)
I see an NBI thread here. First a clarification: it is not uncommon to define cellular automata (CA) as dynamical systems. Some of the concepts carry over: state space, fixed points, periodic orbits, ergodic components, transfer operators, etc. Stating that a CA is a dynamical system with a discrete phase space is an accepted practice. The Wikipedia article avoids that definition because it could create confusion: does retification apply to CAs? what do linear systems have to do with CAs? ...
Exactly - it is not at all uncommon to define CAs as discrete dynamical systems. They fulfil the mathematical axioms of (general) dynamical systems theory. Certainly they are not smooth dynamical systems and (typically) not even metric or topological ones, which restricts what can be said about them in dynamical systems terms. Even so, there are interesting dynamical systems concepts which can be applied to them. These aren't concepts which "carry over" from smooth/metric/topological dynamical systems theory; they are precisely those general dynamical systems concepts which do not require their objects to have topological, etc. structure. 11:37, 24 October 2006 (UTC)
Next, I'll speculate. I have not kept up with the self-organized criticality literature (some exact solutions, use of field theoretical techniques, ergodicity issues, ...), but if I were to approach the problem, I would start out by introducing a dual space. The sandpile model for self-organized criticality is a random cellular automata: you drop a particle, it causes an avalanche, evolving to a fixed point. The process is then repeated. The objective is to understand the statistics of avalanches. In good NBI tradition, introduce a set of observables, functions that associate numbers to the state of the automaton, and consider the operator acting on the functions induced by the time evolution of the CA. The spectrum of this operator should tell you something about the statistics of avalanches. In the space of functions you have a nice smooth (infinite-dimensional) dynamical system with more tools to play with. The difficulty will be that the spectrum will not be nice. Figuring how to control the spectrum will be a result worth publishing. XaosBits 14:49, 15 October 2005 (UTC)
You beat me to the punch about NBI. Linas, I take it you must have spent some time there round-about the same time that Mogens Høgh Jensen and Tomas Bohr were doing their PhD's with PB? :-)
To the wider issue, I'm very glad to see the extensive coverage here of the pure-mathematical field of dynamical systems. I'd just suggest a note in the introduction along the lines of, "Outside mathematics the term tends to be used somewhat more loosely to describe systems of many interacting parts that nevertheless have well-defined collective dynamics." If you like that line, I'll add it to the article, or feel free to do so yourself.
CA in particular I have no axe to grind over, I was just using them as an example. One could also cite, I think, multi-agent systems, swarms, etc. etc. etc., as examples of these loosely defined "dynamical systems". On the subject of SOC, I think much of Deepak Dhar's work follows the lines you suggest (basically transforming the setup into a space of states combined with operators on those states), but though my undergraduate training was in maths, I'm not too knowledgeable about that particular range of mathematical techniques. Turcotte's (1999) review in Rep. Prog. Phys. is a good place to look for references, though I think the analytical literature has grown quite a bit since then.
Nice to have this discussion with both of you, I'll make sure to keep up to date with what you both are doing on Wikipedia. :-) Best wishes, — WebDrake 17:25, 15 October 2005 (UTC)
OK. I guess NBI is Neils Bohr Institute? (and is the NBI tradition considered good or bad?) I did not know any of PB's students. My math education was terrible, causing me to be very disoriented and uncertain. Now that I'm focused on hardcore, nuts-n-bolts math, I feel very revitalized, and much more sure of myself. linas 23:53, 15 October 2005 (UTC)
Yes, Niels Bohr Institute in Copenhagen. A lovely research centre with a grand history (and a grand present!), lots of exciting people working there.

fixed rule -> function (mathematics)?

I'm trying to fix all the pages that link to the disambiguation page for Function. I changed the text "fixed rule" to link to function (mathematics). I hope this preserves the intent of the original author. Volfy 00:22, 23 November 2005 (UTC)

Thank you for fixing it. Fixed rule, I hoped, would be simpler than function in an introductory paragraph. XaosBits 23:32, 23 November 2005 (UTC)
The concept of 'function' is taught to adolescents in basic algebra classes. I think anyone who comes to this page will know what a function is. I also think the phrasing 'fixed rule' seems to imply discrete time. ( (talk) 04:31, 23 September 2009 (UTC))

No time dependent vector fields

If there is a set of equations of the form:

\frac{dy}{d\tau} = u(y, \tau)

then it can be transformed into an equation without time in the right hand side. Consider the equation

 \frac{d}{dt} \left[ \begin{matrix} y \\ \tau \end{matrix} \right]  = \left[ \begin{matrix} u(y,\tau) \\ 1 \end{matrix}\right].

If we set x = [ y , τ ] and v(x) = [ u(y,τ) , 1 ], then the equation in x has no explicit time dependence.   XaosBits 04:16, 27 January 2006 (UTC)

Sure, but this obfuscates features of the field one is often most interested in finding: fixed points and limit cycles. There are never fixed points or limit cycles of the modified system, and proving the existence of the corresponding notions (trajectories asymptotically approaching or spiraling around a line and tending toward infinity) is more difficult.

Sburden 11:11 PM, 21 Aug 2007 (PST)

Redirect from Dynamical systems theory

It looks like that page was redundant so I've redirected it over to here. This is the text from that small article, if anyone feels like integrating it: Elijahmeeks 22:11, 14 February 2006 (UTC)

Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems by employing differential equations.

Within cognitive science, proponents of the dynamical systems theory approach (see dynamicism) believe that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.

External links


I started recreating this article focussing on the theory, rather than the dynamical systems. -- Mdd (talk) 23:39, 7 May 2008 (UTC)

Linear systems and chaos?

I find it quite misleading that chaos theory is mentioned in case of linear flows. The current paragraph observes that there is indeed an exponential change in the distance of two trajectory, but on the one hand in can be an exponential convergence towards each other, and on the other hand it is not typical in the phase space. So, in general there is no typical sensitive dependence on initial conditions in linear systems. Linear systems cannot be chaotic. I would rather remove this sentence. Cumi 09:49, 28 February 2006 (UTC)

The typical case for a linear system is that the separation of trajectories will be exponential (the eigenvalues of A will not be on the unit circle). That does not mean that the behavior will be chaotic. The exponential change in separation is the stretching and shrinking of phase space. The other ingredient for chaos is the folding, and that is missing from linear systems. It states necessary but not sufficient; maybe it could be worded differently? I felt that the point was unusual enough that it should be mentioned, forcing the reader to ponder about the need of the other conditions for chaotic behavior. — XaosBits 12:21, 28 February 2006 (UTC)
I agree, mentioning "necessary but not sufficient" is OK concerning chaos, but before that it is stated that there is a sensitive dependence on initial conditions, which is not the case for all linear systems. Actually, it is never the case for linear systems in the sense it is used in case of chaotic systems. The long-time dynamics is the same for all initial conditions in linear systems: either convergence to the stable fixpoint, or divergence to infinity. Cumi 08:55, 1 March 2006 (UTC)
If sensitive to initial conditions means exponential expansion, then linear systems can certainly display it, but if it means positive topological entropy then linear systems certainly do not. After I wrote the earlier comment I realized that the expression sensitive dependence to initial conditions is often a synonymous to chaos. I use it the way Ruelle has, which is exponential expansion. I changed the text a bit, is it less confusing?  — XaosBits 12:15, 1 March 2006 (UTC)
Yes, my only remaining concern now is that linear systems and chaos are still in the same sentence. What if it is changed like this: Linear systems display sensitive dependence on initial conditions in the case of divergence. Note that in case on nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior. Cumi 13:04, 2 March 2006 (UTC)
Some 'linear' systems can exhibit chaos, eg. a delay differential equation \dot{x(t)}=ax(t)+bx(t-\tau)
As a rule-of-thumb, one can have linear, infinite-dimensional systems; or non-linear, finite-dimensional systems. XaosBits 01:03, 3 August 2006 (UTC)

Definition excludes important cases?

The definition given in the second paragraph seems too narrow. For example it excludes dynamical systems on non-manifold metric spaces and thus all of symbolic dynamics. This may be an echo of the cellular automata thread above, but I wanted to put in a vote for a more expansive definition. If I don't hear an objection I will write a new definition.

More expansive and formal definitions are given later in the article. There is a section that discusses the two main definitions: the one that is geometrical in flavor and one that is measure theoretical. The article is about the geometrical point of view of dynamical systems, and the beginning of the article reflects that. The article is also informal in style. The idea was to have it serve as a guide to other articles in dynamical systems. — XaosBits 00:37, 6 March 2006 (UTC)
OK, that's a fair point about informality. And I'm finding that it's actually very hard to write something that is informal, correct, and embraces both continuous and discrete systems! Come to think of it, is the second paragraph even necessary? The first para gives a nice informal sense already. --Experiment123 01:16, 6 March 2006 (UTC)

No comment link

The external link

was in the wrong section. It also lacked an explanation. XaosBits 22:54, 7 April 2006 (UTC)

Why the "double adjective?"

It would be great if somebody would write up an etymology section for this article. As a non-scientist, I'm having a hard time wrapping my head around why a field of study with this title is considered grammatically correct (in English, at least). The word "dynamic" itself is an adjective; adding the suffix "al" is supposed to turn nouns into adjectives. What purpose is served by making a word an "adjective adjective?" It's like calling a banana a "yellowal fruit." (by on 24 April 2006)

I will speculate. In 1927 George Birkhoff published a very influential book called Dynamical Systems, and the name of the field probably derives from the name of the book. The question is then why would a Harvard professor use the adjective dynamical in dynamical system instead of the adjective dynamic as in dynamic system.
If we look at the Oxford English Dictionary, we can track early usages of the words.
1788 dynamics noun
1812 dynamical adjective
1827 dynamic adjective
1873 dynamic noun
The original usage of dynamics, as the science of moving powers, was meant to include the other sciences, kinetics and statics, associated with mechanical systems. I imagine that it was the word dynamics that was made into an adjective by the addition of the -al, leading to dynamical. So it was not the adjective dynamic that was made into another adjective, but rather the plural noun dynamics. XaosBits 20:46, 24 April 2006 (UTC)
How common is the use of "dynamical" anymore? Here in America, I've always just heard dynamic. LRT24 (talk) 00:36, 27 July 2011 (UTC)
It's common. As far as I can tell, the phrase "dynamic system" is very rare. Jowa fan (talk) 01:20, 27 July 2011 (UTC)
Indeed. Take a look at the references in the Further Reading and External Links sections of the article. I count over a dozen instances of "dynamical systems" in book and web page titles, and only one instance of "dynamic systems". Gandalf61 (talk) 10:31, 27 July 2011 (UTC)

Liouville measure

In the thermodynamic limit there is only one ensemble. Averages computed with different constraints should provide the same result. The Liouville measure μ0 plays multiple roles in statistical mechanics:

  • It is the measure for choosing configurations in statistical mechanics, so the average of an observable a is
 \langle a \rangle = \frac{1}{Z} \int \mu_0(dp,dq) a(p,q) e^{-\beta H(p,q) } .

This role is often ignored in less mathematical texts.

  • It is the measure used to choose the initial conditions the time averages that appear in the ergodic theorems for Hamiltonian systems.

The wording pointed out by Mct mht was unclear about the relation between the measures, so I hope its a little better now. — XaosBits 19:54, 22 May 2006 (UTC)

ok, interesting. Louiville measure often refers to the trivial measure(constant everywhere) on the phase space for the microcanonical ensemble. from what you saying, looks like it refers to the canonical ensemble as well. In physical literature, it seems that  e^{-\beta H(p,q)} d \mu is always taken to be the phase space measure for the canonical ensemble, where μ is the Lebesgue measure, locally. so are you saying, in dynamical systems, μ is not necessarily the Lebesgue measure but its choice is dictated by the ergodicity requirement? Mct mht 20:18, 22 May 2006 (UTC)

I'm not saying anything profound and I agree, μ is the Lebesgue measure of the canonical coordinates (but not any coordinates). Suppose we have a system of non-interacting particles constrained to a sphere and that the energy E of a particle depends on the position of the particle on the sphere. Two angles specify the position: the latitude θ varying from 0 to π and the longitude φ varying from 0 to 2π. The partition function Z for N particles could mistakingly be written as

  Z^{1/N} = \sqrt{\frac{2 \pi}{\beta}} \int_{0}^{\pi} d\theta \int_{0}^{2\pi} d\phi \, e^{-\beta E(\theta, \phi)} .

This integral samples all the configurations and gives them equal weight. The correct calculation weighs the configurations with the Liouville measure

  Z^{1/N} = \sqrt{\frac{2 \pi}{\beta}} \int_{0}^{\pi} d\theta \sin\theta\int_{0}^{2\pi} d\phi \, e^{-\beta E(\theta, \phi)} .

This mistake is hard to make if you start out with a Hamiltonian, but if you start with the energy I could see it happen.

The Liouville measure is the product dp dq where p is the conjugate momentum associated to the position coordinates q. The microcanonical ensemble is given by the averages computed using

 \langle a \rangle_\mathrm{micro} = \frac{1}{\Omega} \int dp dq \, \delta(E - H(p, q)) a(p,q).

in the thermodynamic limit. The dp dq, the Liouville measure, shows up in this average and in the average of the canonical ensemble

 \langle a \rangle_\mathrm{canon} = \frac{1}{Z} \int dp dq \,  a(p,q) e^{-\beta H(p, q)} .

If a long time average is to be equivalent to a measure in space, then the measure needs to be invariant under the flow; and dynamical systems have many invariant measures. In the case of Hamiltonian systems, the Liouville measure seems the right choice, but if the system is not Hamiltonian it's not clear what measure to choose. In the article I wanted to show the analogy between Hamiltonian systems and statistical mechanics on one side and chaotic systems and SRB measures on the other.

The other role of the Liouville measure is choosing initial conditions for the time averages. Some initial conditions do not lead to a well defined average, but those may form a set of measure zero. (This case is a stretch as Liouville measure zero or Lebesgue measure zero are the same thing.) But when the system has multiple phases (up and down spins), the phases may not occur with equal probability, and then the choice of initial conditions matters. — XaosBits 02:15, 24 May 2006 (UTC)

thanks for the above very detailed answer. Mct mht 05:27, 24 May 2006 (UTC)

Measure theoretical definition

"Here, X is a set, and Σ is a topology on X, so that (X,Σ) is a sigma-algebra."

But a topology is hardly a sigma-algebra. When X is a topological space, Σ is usually taken to be the collection of Borel sets. Novwik 19:25, 27 August 2006 (UTC)

Rewrite of 1.10.2006

I rewrote the Definition section in order to highlight the most important definitions and provide a central place for other articles to reference them. I am not exactly sure what definitions should be included in the Definition section at the top. Currently I think we should stick with the geometrical definition on subsets of euclidean space (manifolds tend to distract the lay reader unnecessarily), and move the more abstract measure theory definition to the bottom. I think real dynamical systems and discrete dynamical systems should be discussed at the top but what about holomorphic dynamical systems ? Comments and help appreciated. MathMartin 14:50, 1 October 2006 (UTC)

This is effectively an article about smooth real dynamical systems. Whilst there is certainly enough material in that topic for an article, and they are the sort of system of interest to most scientists, I think that the introduction could indicate that explicitly. It probably doesn't need to do more than say "This article deals with the type of dynamical system typically considered by scientists" or something to the same effect. That allows space for references to articles on other types of dynamical system and should satisfy nitpickers like me. 11:55, 24 October 2006 (UTC)

I tried another rewrite of Definition to highlight the two main types of dynamical systems, namely continuous time dynamical system and discrete time dynamical system, but it did not really turn out too well. Perhaps it would be a good idea to split the article ? MathMartin 19:19, 4 October 2006 (UTC)

I have made some changes to try to keep the article balanced. I removed the excess material on the definition of dynamical systems at the bottom to a separate article along with the new stuff and restored the structure of the article to match the discussion in the overview. Detailed definitions were missing and they deserve their own article.
Dynamical systems is one of the broadest topics in mathematics and one cannot expect a page to do justice to the topic. My hope is that this page be an entry point to other pages.
I did not read MathMartin's comment on manifolds being hard for the lay reader until after I made the changes. Sorry, I may have done things differently. I also noticed that the evolution rule went from f to ϕ. I personally like ϕ for the evolution rule, but I noticed that using Unicode characters in Safari the upper and lower case characters look almost the same, so I decided to use f. XaosBits 00:30, 16 October 2006 (UTC)
Moving the definitions to a separate page was the correct thing to do. I switched the evolution rule to ϕ in order to use f for the underlying vector field of the dynamical system (the discussion of which you deleted !!), but then I encountered browser rendering issues which I thought I had solved by using uppercase ϕ (depending on the browser or os used, there are two distinct renderings of lowercase ϕ, which is confusing for the reader). I agree that dynamical systems is a broad topic and this page should serve as an entry point to other pages, but at the moment there is still too much technical content on the page which is confusing without clear definitions.
Anyway I have noticed your name on a lot of articles on dynamical systems so I thought I would meet you sooner or later :). As I intend to write on this topic in the future I would appreciate you giving my new material a short check for obvious errors. Comments and help are always welcome of course. MathMartin 12:43, 16 October 2006 (UTC)


Hi all, I'm planning on whipping up a template footer for bifurcation theory to sort out the cross-referencing between the different forms of bifurcation. Should such a template perhaps be done at the higher level of dynamical systems? Something that sorted out the assortment of Maps would also be a good addition, though I'm against making too large a template. Maybe I'll look into that after sorting out bifurcations. Feedback welcome! Please discuss at Talk:Bifurcation theory. //jugander (t) 13:00, 17 October 2006 (UTC)


Here is the context: "...he Sultan Catto and his collaborator (Feza Gürsey) exploited internal (dynamical) supersymmetries to construct a combined classification scheme for mesons and baryons."
Yours truly,--Ludvikus 03:31, 16 December 2006 (UTC)
  • In general, the term dynamical denotes the temporal changes of a state. You will find expressions like dynamical behavior or dynamical processes all over the literature on applied mathematics. I assumed in your case, that the term points out the time-dependency of the supersymmetries, i.e. the applied supersymmetries which are not necessarily (1) constant in time. (1: dynamical processes can also be constant.) I hope this answer is useful. DrPhosphorus (talk) 11:35, 9 December 2010 (UTC)

why a paragraph about cognitive theory in a math article?

the entire rest of the article is a very specific area of mathematics. this paragraph seems out of place. Wikiskimmer 07:44, 29 June 2007 (UTC)

yes, indeed. maybe we should find a home for it elsewhere. Mct mht 13:45, 30 June 2007 (UTC)
I sent a note to the person who seems to have added it to the wiki:Aaronp808, to try to find a place for it in the cognitive psych pages...Wikiskimmer 00:05, 6 July 2007 (UTC)

"[...]This new, novel state is progressive, discrete, idiosyncratic and unpredictable." sounds like pseudoscience!

It's not pretending to be science (like astrology, for example) but trying to use DS as a tool to solve problems in cognitive theory. For example, dynamic systems specialists solved the A-not-B_error of young children. I wish I knew more to flesh out the mathematical connections, but I don't. However, I have asked a colleague who does know more to try to expand on this. If he doesn't, then feel free to exile this to another page. --Aaronp808 03:04, 3 October 2007 (UTC)
I created a page for the cognitive science content and moved the related content there. XaosBits (talk) 02:24, 20 December 2007 (UTC)

2nd edition of Abraham-Marsden Book

Foundations of Mechanics is available here as a pdf file. Takes a while to download. MP (talkcontribs) 18:43, 2 July 2008 (UTC)

Clarify the definition?

Although probably correct, the definition is practically incomprehensible for a layman. I would suggest to see the definition and introduction given in the dynamical systems article of Scholarpedia for a more concrete definition. —Preceding unsigned comment added by JuliusCarver (talkcontribs) 11:37, 26 September 2008 (UTC)

Um, you mean "dynamic"?

Does anyone know when the term "dynamical" first arose in English, if it is distinct in any way from the term "dynamic", and if not, why its use is considered acceptable? beefman (talk) 08:09, 26 December 2008 (UTC)

No, they don't mean "dynamic". Dynamic means that something changes or is forceful. Dynamical means that it has to do with the theory of dynamics. In that sense, the OED lists it as dating to 1812. Dan Wylie-Sears 2 (talk) 06:56, 8 February 2009 (UTC)

Etymology? Should that not be included? From Greek dynamikos "powerful," from dynamis "power," from dynasthai "to be able, to have power, be strong enough," (talk) —Preceding undated comment added 03:01, 6 October 2013 (UTC)

Tipping point (climatology)

There's an article Tipping point (climatology), and I'm wondering whether it really is a separate (more restricted?) concept from that of tipping point in the more general theory of dynamics. Here's a definition [1] in the context of climate, and here's the article it's an appendix to [2]. I would appreciate it if an expert on dynamical systems could reply on talk:Tipping point (climatology) or one of the related pages. Dan Wylie-Sears 2 (talk) 07:13, 8 February 2009 (UTC)

Oh, same question about radiative forcing. In both cases I think climatologists doing actual applied math are using concepts from dynamics, and others (not just the general public, but even scientists in related sub-specialties) are mistaking it for a set of concepts unique to climatology. But I don't have the expertise and full-text article access to find sources that would answer the question. Dan Wylie-Sears 2 (talk) 16:26, 8 February 2009 (UTC)


What about some history of the subject? Starting with Poincare until modern developments? Lbertolotti —Preceding unsigned comment added by (talk) 12:08, 19 February 2009 (UTC)

Nonlinear dynamics

Nonlinear dynamics redirects to this page. Wouldn't it be helpful to have a subsection that defines exactly what that means and gives an overview? The reader might infer what it means by reading the linear dynamics section, or might already know that chaotic dynamics is one type of nonlinear dynamics, but a section specifically devoted to summarizing and explaining the importance of the nonlinear case would be helpful.--Rinconsoleao (talk) 08:29, 12 March 2009 (UTC)

Merger proposal

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.
The result of this discussion was to not merge Tiled (talk) 13:35, 27 February 2011 (UTC)

Dynamical systems theory seems to contain either redundant material or info on applications. Tiled (talk) 19:28, 11 September 2010 (UTC)

  • Oppose. I think the reason there are two articles is to avoid edit wars over what the article should contain. XaosBits (talk) 06:49, 4 October 2010 (UTC)
  • Oppose. Given that dynamical systems theory is a major area of math, it seems to deserve its own article, independent of the objects that it studies. This is analogous to the situation with Group (mathematics) and Group theory. Jim.belk (talk) 03:23, 23 November 2010 (UTC)
  • Merge. I would also like to merge in the article Dynamical system (definition). I am not totally opposed to having more than one article on this subject, given how much there is to say on it. However, if we do that it would be good to have a think about how we split the topic up and what information goes in to which article. Yaris678 (talk) 17:07, 26 December 2010 (UTC)
One of the nice things about text on the web is that one can break longer articles into smaller parts giving readers the chance to explore the topic, if the wish, by clicking on links. Because the way contributions are made to the Wikipedia, it is hard to balance an article. One ends up with articles were some points get a lot of importance, giving the casual reader a false sense of the topic. Already there are two technical definitions at the bottom of this article that so far has tried to cover in broad strokes the field of dynamical systems. XaosBits (talk) 20:48, 12 February 2011 (UTC)
  • Oppose. Related, but each sufficiently notable for two articles. Not overly redundant and better treating each separately. Support.and.Defend (talk) 21:27, 19 February 2011 (UTC)

The above discussion is preserved as an archive. Please do not modify it. Subsequent comments should be made in a new section.


I just spent a half-hour doing obviously badly needed cleanups on this article when my browser somehow crashed and all was lost.

Would those editing this article please note that WP:MOSMATH really does exist. Really. It does. Someone worked on this who thought everything included in non-TeX math notation should be italicized, including even digits and parentheses, and used hyphens where en-dashes or minus signs belong, and lots of other stuff like that. Michael Hardy (talk) 15:17, 3 April 2011 (UTC)

Related software

Hello! I saw there's already a section with external links and a friend of mine did a little and interesting iPhone game with some dynamical system concepts. I wonder if it's not worth mentioning. As I'm his friend, think it's better to ask here to avoid problems related to conflict of interest. The game's home page is: Rhalah (talk) 18:31, 30 June 2012 (UTC)

Creative Commons License