Talk:Attractor/Archive 1

Archive 1

Shed some light

Can anyone shed some light on why a strange attractor is the most common type of attractor? It seems to me that periodic attractors are at least as common (whatever 'common' means) as strange attractors (bifurcation and all that), if not more so. And, although it was before my time, apparently most mathematicians used to think that all attractors were periodic, hence 'strange attractors' -- so what's the story? Ben Cairns 05:36, 10 Dec 2003 (UTC)

Initial misconceptions

"The type of attractor exhibited by this phenomena is known as a 'point attractor', because the limit set consists of a single point:"

This is in a one-dimensional universe, right? It seems to me that the final state of the book would tend to be anywhere in the plane of the floor for a three-dimensional universe. Is there a better example? also y or h is typically associated with height, not x. I don't know much about the concept of attractors though. - Omegatron 21:19, Aug 4, 2004 (UTC)

"A planet orbiting around a star is an example of a periodic attractor."

Planets don't "fall into" orbit though. if you just stuck a planet with no initial velocity near a star it would just fall into the star. Disturb an orbiting planet from its orbit and it doesn't slowly tend back into the same orbit. Tidal locking seems more like an attractor to me than orbiting. - Omegatron 21:24, Aug 4, 2004 (UTC)

point attractor - in that example, it is a one dimensional universe. In dynamics, the variables start at x, without regard to physical analogy. That's why x is used instead of y. another example might be a damped pendulum: it will eventually stop swinging.

limit cycle/periodic attractor - I don't know much about tidal locking, but it sounds like it would be a good example. Another example would be microphone feedback, since the a single tone; oscillation (albiet w/harmonics, it is still periodic with period of the base harmonic) is amplified until the amplifier saturates, and is then a fixed amplitude oscillation. after a small increase or decrease in the amplitude, it will quickly return to the attracted amplitude.

In any case, it seems like you have a good intuitive grasp of the concepts. Kevin Baas | talk 00:14, 2004 Aug 6 (UTC)

A strange attractor is a non-periodic attractor. This is the most common type of (not spatially-extended)... Sorry if I got this wrong, but isn't the Lorenz attractor supposed to be 3D? His differencial system has 3 variables to be associated with 3 directions of phace space… Or not? Koolniczka

right. phase space is not to be confused with physical space. Kevin Baas^talk 23:45, 2005 Apr 25 (UTC)

Well.. then attractor itself is not 3D or..? I though about an attractor as a geometrical representation. If it has more than 2 dimensions, then it should not be described as not spatially-extended. Actually IIRC, it has some fractal dimension (2.4 or smthn like that ;-) so I'm aware that this could run tricky, calling it 3D but it certainly looks to me as spacialy extended.
ANTIFLAME NOTE: I'm merely an informend layman. So if I'm completely wrong, just tell me an' I'll go RTFM ;-) Koolniczka 00:08, 26 Apr 2005 (UTC)

the antiflame note is unneccessary, my wiki-stress-level is 0, and you're very civil. phase-space is a.k.a. state-space. each point in phase space represents a possible state for a system, for instance, 3 phase space dimensions, x,y,z, x = 5, y = 2, z = 4, could represent a temperature of 5 units, pressure of 2 units, and volume change of 4 units. or they could represent changes in the populations of rabbits, fox, and mice. the mice are 3-d, for sure, but the attractor refers to a set of states that do not take up any physical volume, (unless you're discussing morphogenesis, in which the attractors would actually be the mice, so to speak.) the phase space embeds the attractor. the dimension of the attractor is much less than that of the embedding phase space. for instance, a strange attractor embedded in 3 dimensions has less than 2 dimensions. The attractor has no extension, though; it exists at a single point. each dimension of the phase space is essentially a different timeline, not a different spatial continuum. to demonstrate this, it can be shown that actually only one variable is needed, and the attractor can be deduced: plot the first time derivative of that variable on one axis (the velocity), the second on another (the acceleration), and the third on yet another (the jerk). all the information can be contained in a single, analog signal, and sent over, for example, a copper wire (up to the information carrying capacity of the wire).

If, however, an uncountable set of such systems, independant of each other, constituted a composite system, to transmit the state of that system would require a copper wire for each constituent system: an uncountable set of copper wires - more theoretically, information channels - and therefore the composite information channel would have to have (lateral) spatial extension to transmit the state.

also, this would amount to an uncountable number of phase spaces, or one big composite phase space with an uncountable number of dimensions. so that's the difference between spatial extension and lack thereof in the sense employed: finite dimensional phase space and uncountable/infinite dimensional phase space, respectfully. the difference between a dimension of phase space and the space meant, is: for the space meant, each point has something "in" it, that has unique information; is independant of the other points. for a dimension of phase space, only points that are part of a certain trajectory have something in them, and only that trajectory - or more accurately, there is a vector field defined over the space, which can be reduced to the selection of a few coefficients in the equation of that field; a few points on a few number lines. in the space meant, for a one-dimensional extension, for each point on a line there is a value, and thus there are an infinite number of values, as opposed to a point on a number line, where there is only one value.

sorry if this is too much. Kevin Baas^{talk: new} 00:51, 2005 Apr 26 (UTC)

An attractor is a set of related rates of change in a system that are asymptotically approached. each dimension in "phase space" is such a rate. Don't be confused by the use of the word "space" into thinking that the attracto has extension. An attractor is still a set of related rates, and the attractor is only spatially extended if those rates are spatially extended. For instance, if x₀ refers to a rate of change of a variable at a point, x₁, refers to a rate of change of a variable at a point 1 cm from that point, and the variables x_0<R<1 refers to the points inbetween, one has a spatially extended dynamical system. the phase-space would have a dimension for each real number between (and including) zero and one. Kevin Baas^{talk: new} 01:10, 2005 Apr 26 (UTC)

Kevin Baas^{talk: new} 01:06, 2005 Apr 26 (UTC)

Perhaps this is more clear: the state of a non-spatially extended system, at any given point in time ("instant"), can be expressed as a single point, whereas the state of a spatially extended system, cannot. So to the system, to the attractor. The dimensions in phase space, again, correspond to derivatives w/respect to time, not extensions of space. Kevin Baas^{talk: new} 02:40, 2005 Apr 26 (UTC)

But as soon as the dimensions in a phase space become uncountable, one needs an extension of space to "decompose" the phase space into an uncountable set of finite-dimensional phase spaces. If the state of each phase space is represented by a Real number R, one requires an uncountable set of Real numbers to represent the state of such a system. Not to be confused with "a selection from an uncountable set of numbers", which could just be one Real number. Kevin Baas^{talk: new} 05:01, 2005 May 9 (UTC)

An attractor is a set of related rates of change in a system that are asymptotically approached. each dimension in "phase space" is such a rate. Don't be confused by the use of the word "space" into thinking that the attracto has extension. An attractor is still a set of related rates, and the attractor is only spatially extended if those rates are spatially extended. For instance, if x₀ refers to a rate of change of a variable at a point, x₁, refers to a rate of change of a variable at a point 1 cm from that point, and the variables x_0<R<1 refers to the points inbetween, one has a spatially extended dynamical system. the phase-space would have a dimension for each real number between (and including) zero and one. Kevin Baas^{talk: new} 01:10, 2005 Apr 26 (UTC)

Kevin Baas^{talk: new} 01:06, 2005 Apr 26 (UTC)

Perhaps this is more clear: the state of a non-spatially extended system, at any given point in time ("instant"), can be expressed as a single point, whereas the state of a spatially extended system, cannot. So to the system, to the attractor. The dimensions in phase space, again, correspond to derivatives w/respect to time, not extensions of space. Kevin Baas^{talk: new} 02:40, 2005 Apr 26 (UTC)

Thanx for reply and patience ;-) Hope you keep it ;-)...
>>...a strange attractor embedded in 3 dimensions has less than 2 dimensions..
I cannot resist to oppose your statement here ;-). As I found out (sepecifically speaking at http://hennequi.pierre.club.fr/Page%20vierge%206.htm, bottom of the page), the dimension of Lorenz attractor embedded in three dimensions of phase space (as it has three degrees of freedom) is approximately 2.062. You could surely reduce the degree of freedom by one and have a lesser dimension but as an attractor is (AFAIK) allways a representation of the behaviour of a specific system, this representation should rely on the nubmer of the degrees of freedom the given system has and therefore respect the given number of dimensions. (And once more, I'm not claiming my interpretation is correct, I'm just presenting my reasononing)
>>An attractor is a set of related rates of change in a system that are asymptotically approached. ... The dimensions in phase space, again, correspond to derivatives w/respect to time Yes, I would say, my common sense is satisfied with this explanation. But then as a set of data it can be (theoretically speaking) made a transformation upon, and the representation can be (I assume) seen and treated as a geometrical structure.
So (for to come to the article at last) the division of attractors (point attractors, periodic point attractors, periodic attractors, strange attractors, and spatial attractors in the head of the article) implies (to me at least) that a geometrical approach to attractor is relevant. Hence I would propose (and humbly admitting not being able of doing it myself) to clarify the difference between space attractors and others (especially the strange ones ;-) ). And subsequently I wold encourage a wider distinction between periodic point attractors and periodic atractors (I would prefer to distinct them as periodic discrete attractors and periodic continuous attractors, though I'm not sure if this is correct). Another point that I'm uncomfortabe with is the "infinite loop" example for both of them. I find this rather confusing.
Well... I hope my reasoning is not to confused and wish god luck to those who read it ;-)
Koolniczka 16:21, 26 Apr 2005 (UTC)

well spoken. i agree that discrete and continuous attractors should be distinguished explicitly. I agree also that attractors can be examined geometrically. perhaps its a philosophy point involved when i say that strange attractors still dont have "extension". Kevin Baas^{talk: new} 17:14, 2005 Apr 26 (UTC)

Spatial attractors

Could someone illuminate the spatial attractor section? An attractor is a collection of points of the phase space of a dynamical system (or its closure, depending on how one wnats to define it) towards which a large measure of initial points converges to. Does spatial attractor fit this definition? What is a pseudo-example? Is there a reference? XaosBits 20:26, 30 Apr 2005 (UTC)

a spatial attractor is where the attractors are combinations of points across a continuum of phase spaces over a spatial continuum. Examples are Turing structures and emergent media. Kevin Baas^{talk: new} 21:28, 2005 Apr 30 (UTC)

I was bold. I hope I wasn't reckless. The point I was trying to clear up were the differences between limit sets, invariant sets and attractors. I also imagine that what is meant by spatial attractor is an attractor of a partial differential equation. The article needs a discussion and some explanations on that point.

You didn't understand me, then. Another example of spatial attractors is morphogenesis. With regard to living things (as morphogenesis has come to traditionally be used) the attractor is an organism. And I don't mean this is the purely intuitive sense. I mean this mathematically. Spatial attractors can be discovered in excitable media (sorry, i said emergent before, i meant excitable) and reaction-diffusion systems. Kevin Baas^{talk: new} 23:39, 2005 May 1 (UTC)

The biggest cut was with regards to attractors for discrete dynamical systems (cellular automata and family). Very little of the theory of flows extends to cellular automata and most authors exclude cellular automata from the definition of dynamical systems by requiring that the phase space be a manifold. I thought the Wikipedia could follow the tradition. XaosBits 20:04, 1 May 2005 (UTC)

The phase space of a discrete dynamical system is a 0-dimensional manifold (i.e. a finite set of points). Kevin Baas^{talk: new} 23:39, 2005 May 1 (UTC)

I agree on the fact that spatial attractors needed further explanation (or deletion as it seems ;-).But I feel a bit unconfortable with the proposed division of attractors ( again ;-) ). As Kevin Baas advised me above, attractors only exist in phase space (wich, as I understand it, is a geometric construct). The division on behalf of "dimension", with respect to the number of the degrees of freedom the given system has, could be more rigid.
I'd like to propose this as RFC:
1.attractors of integer dimension
(posible subdivision ad infinitum -special case A) dimension 1 (discrete attractor - point)
-special case B-I) dimension 2 (discrete attractor -points)
-special case B-II) dimension 2 (continuous attractor - curve)
-special case B-III) dimension 2 (contimuous attractor - surface)
-... )
2.attractors of fractal dimension a.k.a. Strange attractors
Anyway, the point I'm tryin' to get to here is to firmly distinguish "strange" and "non-strange" attractors on behalf of their "behaviour". "Non-strange" attractors are predictable finite sets of points in phase space (corresponging to a finite set of d(x,y,...)/t(x,y,...) of the given system) and they represent a nonchaotic behaviour. As opposed to "strange" attractors wich are infinite sets (if im getting this right,... well at least Lorenz attractor is a infinite "spacial" curve confined in a finite amount of "space") and represent chaotic behaviour (infinite set of d(x,y,...)/t(x,y,...) ).
And now for something completely different... objection yor'onnor! ;-)
With regard to living things (as morphogenesis has come to traditionally be used) the attractor is an organism. And I don't mean this is the purely intuitive sense. I mean this mathematically.
Looking at these words and the discussion (and the article as well) I would beg for less ambiguous examples of attractors, namely the distinction between an attractor and the system it is "pulled out" of. I would say that attractors can be rather extracted (deduced) from the phenomena than the phenomena beeing attractors themselves.
P.S. If someone would have the impression that I just rant and do nothing about the article itself, it's not that I'm lazy, it's rather that I don't dare to touch it without some sort of expert aproval. Objection of beeing too much talkative towards the things I don't really understand could be more justified but... hmm... I just can not resist ;-) In the end, encyclopedia should be readable by the laymen; treat me as a kind of chaotic feedback. Koolniczka 23:02, 3 May 2005 (UTC)

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It would be interesting to introduce a nomenclature such as the one Koolniczka is suggesting: it's logical and would help classify the attractors that are found in systems. But as fas as I can tell, it is not common in the dynamical systems or chaos literature. I did a little (un-scientific) usage experiment: I searched on 3 May 05 a few of the terms used in the attractor page on Google and Google scholar and reported the number of hits.

term	Google	Google scholar
strange attractor	153000	4300
limit cycle	92200	18800
periodic attractor	751	867
point attractor	6750	1100
fixed point	1180000	324000

The terms strange attractor and limit cycle are common, but periodic attractor and point attractor are rare. Instead, the terms limit cycle and fixed point are used.

It is possible to have a chaotic dynamics with an attractor with integral dimension. Take a two-torus, give it a vector field corresponding to geodesic flow, and then add an extra dimension that contracts to the two-torus. The two-torus is an attractor and the dynamics is chaotic.

Feedback is always appreciated. Maybe the way to approach the attractor classification issue is to follow the historical route. There were fixed points and limit cycles. I'm sure if asked about it, a mathematician in the 1950s would say: "Yes, a torus can be an attractor," but then it wasn't a helpful concept. It was not common to think of a dynamics settling onto an attractor. It is only after the Ruelle and Takens paper that the idea started catching on. XaosBits 01:44, 4 May 2005 (UTC)

I think more accurately, a torus would be a quasi-periodic attractor. Though it's sensitive to perturbations in that it is hamiltonian in at least one dimension, it is different than a strange attractor in that it doesn't produce information; it doesn't have "chaotic dynamics" when it is not perturbed, like a strange attractor does (chaotic dynamics are intrinsic), that is, it doesn't have "chaotic dynamics". It does not amplify noise. it's unpredictability doesn't increase exponentially with time, it increases with cumulative external perturbation. That's hamiltonian, not chaotic. Kevin Baas^{talk: new} 02:33, 2005 May 4 (UTC)

By two-torus I meant the double donut, the double torus, the genus two two-dimensional surface (easier drawn than described). Sorry for the confusion. The dynamics is similar to the Sinai billiard. XaosBits 13:08, 4 May 2005 (UTC)

Admittedly, I'm not familiar with that. In any case, if it has a homoclinic point, or if in every metric the prediction accuracy has a minimum decay rate that is irrespective of the frequency and amplitude of external perturbations (e.g. extrinsic "noise"), i.e. if the equatiosn have a positive lypanuv exponent, that is, if the system produces information (with respect to time), then I agree that it has chaotic dynamics. If not, then I say it has at best hamiltonian dynamics. Again, I don't know enough to determine.

In any case, I can't think of any reason why choatic attractors would be limited to integral dimensions. The only limitations I can see is that an attracting set must have a lower dimension than the phase space in which it resides. (or if the set is multifractal, the highest dimension in the multifractal must be lower than that of the phase space), that and a strange attractor must reside in a phase space with dimension greater than 2. Kevin Baas^{talk: new} 06:30, 2005 May 7 (UTC)

>>...a strange attractor must reside in a phase space with dimension greater than 2
Why so? My nonmathematical mind can comfortably imagine a chaotic attractor embedded in 1 dimension (some Cantor-like set, finite lenght, infinite amount of points). In fact, I think that logistic equation could produce such an attractor.
>> I think more accurately, a torus would be a quasi-periodic attractor...it doesn't have "chaotic dynamics" when it is not perturbed, like a strange attractor does (chaotic dynamics are intrinsic)...
This is a good point. That's basically what I was trying to introduce above; chaotic dynamics create a "non-smooth" attractors, non-chaotic attractors are well defined, they can be descibed by Euclidean geometry. Chaotic attractros are basicaly the structures of infinite complexity ( they produce information as Kevin Baas wrote) and therefore unrepresentable.
And concerning my pseudotaxonomy up there, it is more intuitive than exactly defined and it really doesn't honnor any conventional nomenclature (as XaosBits mentioned). My initial point was: integral dimension=non-chaotic, fractal dimension=chaotic. Koolniczka 23:53, 8 May 2005 (UTC)

By "reside in" a phase space of dimension greater than 2, i did not mean to imply that the dimension of the attracting set had to be greater than 2, i meant that the phase space must have more than 2 degrees of freedom in order to embed an attractor that is both bounded and non-periodic.

I understand the suggestion that in order for neighboring points to diverge exponentially, the attracting set must be non-smooth. I know that, as a force vector field, the phase space must have positive divergence in at least one degree of freedom and negative divergence in at least one on every point of the attractor for the attractor to be strange. I can't see right now if this neccessarily leads to a manifold that is not smooth at any point, and I can't see right now if such a manifold is neccessarily fractal. Yet there might be some relation of the exponential divergence and convergence to the exponential scaling law of a fractal. Thou it's important to remember that an integer-dimensional form is just a special case of a fractal-dimensional form. In any case, not knowing what XaosBits refers to in his 2-torus example, I'm not convinced that he is refered to a true strange attractor. Kevin Baas^{talk: new} 01:32, 2005 May 9 (UTC)

I noticed on the Chaos theory the statement: "Strange attractors have fractal structure." If true, the "or" in "An attractor is informally described as strange if it has non-integer dimension or if dynamics on the attractor is chaotic." should be a implication: if it is a strange attractor then it has fractal structure. (but non neccesarily non-integer dimension? what is meant by "fractal structure"?) Kevin Baas^{talk: new} 03:26, 2005 May 9 (UTC)

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(Rather than indent further, lets bring it back to the margin XaosBits)
A few points:

1. A flow on a torus cannot be chaotic.
2. A ODE of two variables defined on the plane cannot be chaotic.
3. A map in one dimension can be chaotic.
4. The double donut flow I mentioned is the Sinai billiard. This is a famous example in dynamical systems, as it was the first example of a realistic looking system that was shown to be ergodic. Sinai considered two disks bouncing around a square with periodic boundary conditions. The other famous billiard model in dynamical systems is the Bunimovich stadium.
5. One can have chaotic dynamics with smooth attractors. The variant (earlier on) of the Sinai billiard is an example. The Baker's map with an extra dimension that contracts to zero would be another example. That is, imagine {B_x(x,y), B_y(x,y)} is the Baker's map. Then create the map C(x,y,z) = {B_x(x,y), B_y(x,y), 0.3 z}. The attractor for this map is the square [0,1]².
6. A strange attractor was defined informally by Ruelle and Takens in their 1971 paper as an attractor with the structure of Cantor set in one dimension. In Ruellle's book on dynamical systems, he defines a strange attractor as an attractor on which the restriction of the dynamics is chaotic. There is no relation between an attractor being fractal dimensional and the dynamics on it being chaotic. The term strange for an attractor is an informal term meaning that the attractor is either fractal or the restriction of the dynamics to the attractor is chaotic. I have modified the article accordingly.

XaosBits 03:32, 9 May 2005 (UTC)

A map, yes, discrete time. But physically discrete time is actually time as counted by a dynamical system with a singularity, such as a flip-flop circuit (which might use, for instance, transistors operating in the saturation regime to produce a limit cycle with periodic singularities which may, with usually negligible information loss, be conceptually reduced to a discrete dynamical system). One can construct a simpler physically possible model: take a Pulse_computer, and use the spike interval time as "x" and the spike count as (discrete) time. Doing this with one neuron in a network, one constructs a logistic map. However, for the map to be chaotic, the neuron (or neurons) has (have) to have more than two interdependantly varying potentials.

My point is that lower-dimensional systems like logistic maps are conceptually reduced versions of higher-dimensional dynamics, which are reducable on account of singularities. And that discrete-time systems with chaotic behavior, as reductions of higher-dimensional systems, can only be reductions of continuous-time dynamics w/more than 2 degrees of freedom. Kevin Baas^{talk: new} 04:45, 2005 May 9 (UTC)

And in any case I don't know if a logistic map is considered a "strange attractor". From my knowledge, "strange attractors" are, by definition, processes in continuous-time dynamical systems. Kevin Baas^{talk: new} 04:06, 2005 May 9 (UTC)

And in any case, a logistic map is algebraic, not differential, and thus lacks a true phase space. Kevin Baas^{talk: new} 04:11, 2005 May 9 (UTC)

If a strange attractor is, by definition, aperiodic and bounded, then there must be at least one finite-length curve in phase space that intersects the attractor an uncountable number of times (to not intersect somewhere new just one more time would be to repeat; to be periodic) - as distinct from a finite number of times or one long contiguous time. Therefore the dimension of intesection of the strange attractor with that curve is greater than zero and less than the one (the dimension of the curve, and I'm refering to a 1-D curve). "Fractal structure". Kevin Baas^{talk: new} 05:29, 2005 May 9 (UTC)

more specifically, I didn't clarify: it is possible for the curve to intersect the attractor for a long continuous time, and still intersect an uncountable number of other times. As well it is possible for it to intersect contiguously a couple of times, and still intersect an uncountable number of separate times. However, it is not possible for it to intersect contiguously an uncountable number of times, because that would result in a dimension of intersection greater than the dimension of the curve. In any case, the set of combined "enter" and "leave" points of the attractor on the curve must have a (hausdarf) dimension less than that of the curve (1), and more than zero (in order for the attractor to be aperiodic). Kevin Baas^{talk: new} 05:41, 2005 May 9 (UTC)

Same with a plane curve: at most an the attractor can intersect an uncountable set of lines (distinct from points), so long as the dimensions of the uncountable set is less than one, and with larger-dimensional intersections the sum of the intersection dimension and the dimension of the set of such intersections must be less than two (the dimension of the plane curve), and therefore the dimension of the set must be less than one. with points, picture a mesh of points of any pattern. for the points to be uncountable without turning into lines, that mesh must be scaled and duplicated fractally. (if it was scaled finitely, the points would still be finite in number. if it was scaled infinitely, the points would turn into lines or a plane, or some other dimensionality form depending on the R^x of the infinity (by "R^x of the infinity" I mean whether it is scaled infinitely in 1 dimension (R¹), 2 dimensions (R²), or somewhere inbetween.) The same logic applies to higher-dimensional curves. Kevin Baas^{talk: new} 06:13, 2005 May 9 (UTC)

"A Riemann surface of genus g is essentially a two-torus with g holes instead of just one." - is this what you refer to? Kevin Baas^{talk: new} 06:28, 2005 May 9 (UTC)

>>Thou it's important to remember that an integer-dimensional form is just a special case of a fractal-dimensional form.
Fractal dimension as I understand it is a mathematical concept of dealing with structures of infinite complexity, I realise that this notion has little in common with Euclidean dimension. Maybe it was misleading (or I was misleaded myself;-) )to mention integral dimension as opposed to fractal dimension. I'm inclined to believe that the notion of "fractal dimension" (as a mathematical concept) can be applied to any structure. Then "fractal dimension" of a structure can be either na integer or a fraction. Right?

That's what I mean. By "fractal" is not neccessarily meant self-similiar. By "fractal" is meant that the dimension doesn't have to be a natural number, but can be arbitrary, (thou generally not negative. perhaps when the application of fractional calculus to fractals becomes more common, along with the paradigm of information theory, the idea of negative dimensions will become more natural), which includes the set of real numbers. So technically speaking, insofar as 1/1 is a fraction, fractals can have integer dimension. But typically the word has been used to refer to non-integer dimension. Kevin Baas^{talk: new} 16:59, 2005 May 9 (UTC)

>>My point is that lower-dimensional systems like logistic maps are conceptually reduced versions of higher-dimensional dynamics,..
Basically, I would agree. But a system, as I understand it, is a mathematic set of formal rules rather than a real("material") thing. Therefore I would say that reducing the system by deliberately reducing the degree of freedom, doesn't really matter. We will have two sets of formal rules, see two systems, allthough they may be related in the "real world".
>>And in any case I don't know if a logistic map is considered a "strange attractor". From my knowledge, "strange attractors" are, by definition, processes in continuous-time dynamical systems. And in any case, a logistic map is algebraic, not differential, and thus lacks a true phase space.
Hmm... I really dont have the qualification to contest that. Basically what I had in mind: Have a logistic equation, it is a series defined by recurrence. Section of the logistic map for a parameter which exhibits chaotic behaviour is an unlimited set of points comprised in a finite lenght.Though I'm not sure now if it is formally correct to mention this section as an attractor in a phase space of dimesion 1. >>In any case, the set of combined "enter" and "leave" points of the attractor on the curve must have a (hausdarf) dimension less than that of the curve (1), and more than zero (in order for the attractor to be aperiodic). This is better deffinition.
Anyhow, some questions at last:>>By "reside in" a phase space of dimension greater than 2, i did not mean to imply that the dimension of the attracting set had to be greater than 2, i meant that the phase space must have more than 2 degrees of freedom in order to embed an attractor that is both bounded and non-periodic. My question then would be: Is'it possible to ennumerate the degree of freedom for the logistic equation? What's the difference (or relation) between the chaos in differencially described system and the chaos in iteratively described system?

well technically a differecially described system is also recursively described (ergodic theory), so in that way the two are similiar - the difference is basically continuous time vs. discrete time. And what I said above is that discrete time is neccessarily a conceptual compression of a continuous-time attractor with temporal singularites, so that's how the two are related. They both may be chaotic, and as far as i can tell the chaos is basically the same except it happens in "spurts" in discrete time. What I'm suggesting is that it may be misguided to use a logistic map as a counter-example of "a strange attractor must have more than 2 dimensions." perhaps this is where distinction between discrete and continuous time may be useful in the article - if logistic maps belong in the article. And I would say they do, but not as a strange attractor. Ofcourse, I would also say that spatial attractors belong in the article. ;) Kevin Baas^{talk: new} 16:59, 2005 May 9 (UTC)

I agree that it was a false step to use this LE example. My fault, I was wrong. Now I think I can see what you meant. Well, to sum it up: strange attractor must be embeded in a phase space of topological dimension >=2; strange attractor can be of fractal (aka Hausdorff-Besicovitch) dimension < 1. Right?
Koolniczka 22:48, 9 May 2005 (UTC)&Koolniczka 22:54, 9 May 2005 (UTC)(corrected myself)Koolniczka 16:43, 10 May 2005 (UTC)(1oftoomany ;-))

in the above i was refering to the intersection of a strange attractor and a certain hypothetical line, in attempt to show that strange attractors do neccessarily have "fractal structure". (Thou I admit there is a weak point in my proof: the point intersections on plane curves, I don't know if its neccessary that they be fractally distributed like that, i just couldn't picture it any other way.) A strange attractor, being in continuous time, must have a dimension of at least one, because, over a finite interval of time, it must have a unique state (remember it's non-periodic) for each infinitesimal moment of time, and thus it must have at least as many "dimensions" (degrees of freedom/parameterizing variables) as "time": one. At least one. Now a discrete-time system might theoretically have an attractor with dimension less than 1, since the "dimension" of time is zero. But I wouldn't call any discrete-time system, even a chaotic one, a strange attractor. When "strange attractor" was coined it was in reference to a continuous time system, and I've never heard it used for a non-continuous time system, though I don't know if this delineation is "official", so to speak. Kevin Baas^{talk: new} 05:53, 2005 May 10 (UTC)

although a logistic map doesn't have enough dimensions to have simultaneous divergence and convergence- pehaps a difference is that it alternates between divergence and convergence? Kevin Baas^{talk: new} 17:17, 2005 May 9 (UTC)

Ad XaosBits:
>>2. A ODE of two variables defined on the plane cannot be chaotic.
5. One can have chaotic dynamics with smooth attractors.
Could you (or someone else) digg deeper into this. I'm not confortable with these statements. I would appreciate some more (but rather simplistic ;-) ) explanation.
Koolniczka 15:14, 9 May 2005 (UTC)

A ODE in two variables defined on the plane cannot be chaotic is an informal shorthand for the Poincaré-Bendixson theorem. Sometimes the statement that there is no chaos in two-dimensional flows is taken to mean Peixoto's result on structurally stable vector fields on two-dimensional compact surfaces.

The Poincaré-Bendixon theorem says that if you have a system of the type:

${\dot {x}}=u(x,y)$

${\dot {y}}=v(x,y)$

then as long as the {u,v} vector field is smooth and has a finite number of equilibrium points, the dynamics of the vector {x,y} cannot be chaotic.

Another way of looking at the double donut example is to think of a Hamiltonian chaotic system where energy is conserved. You had 2n variables, and with the extra constraint, 2n - 1 variables. The sub-manifold M implied by the constraint is not a fractal, as the 2n original one was not. Now create a new fake system, not necessarily Hamiltonian, that has an extra dimension and converges to the M manifold. The 2n-1 dimensional manifold M is now an attractor in the fake system.

But "Hamiltonian chaotic" is an oxymoron. Also, if energy is conserved, then the system would be reversible, and if it's reversible, then it's not chaotic. (In a strange attractor energy is not conserved, it is simultaneoulsy absorbed and dissipated. There are multiple meanings of the word "chaotic" and let me be clear here: I mean that it is not a strange attractor. (where's the positive lyaponuv exponent?) Kevin Baas^{talk: new} 05:31, 2005 May 10 (UTC)

The chaotic × non-chaotic and fractal × non-fractal aspects of an attractor have been often discussed. See question 2.12 of the FAQ for sci.nonlinear.

I hope some of these buzz words help you find a good reference. Some are listed the chaos page.  XaosBits 02:54, 10 May 2005 (UTC)

(break)

Thanx for toll-free education to both gentlemen ;-) Well, recap, just to be sure.
1. No chaos in continuous systems representable in phase space of topological dimension <3.
2. Therefore in continuous systems no strange attractor of Hausdorff-Besicovitch dimension <2 .
3. No (formal) strange attractors in recurrently defined discrete systems.
One question remains: Can we have "real" chaos with smooth attractors?
Koolniczka 16:43, 10 May 2005 (UTC)

1.) I would change the not <3 to not <=2. It might be limited, as you say, to not <3, but we don't know that. (unless XaosBits has any insight into that, or that FAQ he mentioned) We only know that it must be >2, because there is no way to draw a line of infinite length (that is not a circle; not periodic) in a bounded space in 2 or less dimensions. (Note: this is a topological argument.)

2.) Admittedly, I don't know if strange attractors have to have hausdorff dimension > 2, I only know that they must be > 1. But I can't say for certain, and I can't present any counterexample.

-) Google said

Rossler --> estimated between 2.01 and 2.02; Lorenz --> approximately 2.062Koolniczka 18:16, 10 May 2005 (UTC)

question.) By "real" I assume you mean with at least one positive and one negative lypanov exponent? (i.e. produces and destroys information "intrinsicly") Off hand, I don't see why not, if you mean by "smooth" contiguously varying through time. (Certainly one can always find an arbitrary curve wich intersects a form, attractor or not, erratically/discontinuously.) I know there is such thing as intermittent chaos, where there are periods of what looks very much like white noise in the time series, though i've never heard anything to the effect of this being universal.

To clarify things: as I my reasoning goes, "smooth" attractors are well defined (simple) topological structures (or can be treated as such), they have finite boundaries. As opposed to strange attractors which are infinite (one way or another) and produce information (you can watch them unfolding for two eternities and they never repeat). And Kevin Baas (AFAICS) tried to challenge the example of "smooth" though chaotic attractor mentioned by XaosBits. Koolniczka 18:16, 10 May 2005 (UTC)

Well my confusion with "finite boundaries" is thus: in the true dimension of a boundary, a boundary may be finite. But in a higher dimension, such a boundary is infinitesimal, and in a lower dimension it is infinite. For example, a finite-area plane has infinite length and no volume. So the first question would be: what is the dimension of the boundary?

By "smooth" I think "vary contiguously" - that is, in the fourier (frequency; sine/cosine decomposition) domain, the "power"(amplitude) vanishes as the frequency approaches infinity. i.e. each point in a function has a neighborhood of points in that function that are arbitrarily close (infinitesimal distance), and the dimension of that neighborhood is equal to the dimension of the range of the function.

However, I'm kinda fuzzy on the whole official topological definition of "smooth", either that or the definition needs work, because I have the impression that the current definition classifies all non-integer dimensional forms as "non-smooth". Is it that the definition is not trully intrinsic (like tensor calculus)?, that it doesn't fully embrace the idea of non-integer dimensions? Another possible defintion of smooth: has a derivative at all points. Has how many derivatives? (C^? smooth?) It has been shown by way of fractional calculus that there are fractals (such as the weierstrass function) that can be differentiated, it's just that the derivatives are of non-integer order. Kevin Baas^{talk: new} 18:44, 2005 May 10 (UTC)

Layman's definition of smooth ;-) :"varies contiguously" and "finite in any dimension lesser than the dimension of space it is embedded in" (as we speak of phase space) or "basicaly of integral Hausdorff dimension" ;-), e.g.: circle, (closed) spatial curve (of any shape), suface of the sphere, any finite surface (of any shape), surface of the conus, surface of the torus... Sorry but I lack the notion apparatus to describe it mathematically correct.Koolniczka 19:37, 10 May 2005 (UTC)

You got the inequality backwards there: a surface has infinite (unbounded/undefined) length - so you mean finite (or zero; infinitesimal) in any dimension greater than the dimension of the attractor. Well if it has to be aperiodic, then it has to be infinite in at least one parameterization (degree of freedom): time, and therefore if it were a line, it would have infinite length, if it was an 2-d form, it would have (possibly finite width times definitely infinite length equals) infinite area, etc. if it were a circle, it wouldn't be a strange attractor. regards "varies contiguously", i attempted to tell you what i know above by saying that, in regards to one degree of freedom: time: the only counterexample of varying contiguously w/respect to time I can think of is intermittent chaos, and I don't think that's universal (present in all strange attractors). And I don't know if it varies contiguously in all directions, and the fact that it may be of non-integral dimension complicates matters: you said you don't consider it smooth if it is of non-integral dimension - but it's concievable that a form may vary contiguously w/in a fractal dimension. (for example, the weierstrass function is fractal and has a finite (fractional) derivative at all points if it is differentiated to the degree of it's dimension. (i.e. if it's dimension is 2.34, one takes the 2.34th derivative and finds a finite solution at all points, and it therefore contiguously varies through 2.34-space.). Maybe I'm the wrong person to ask. But I will say that I think my "curve" proof of "fractal structure" may just be a "poincare section" - and that I was attempting to show that a strange attractor must impress a fractal on a poincare section, because it must, by definition, be aperiodic and bounded. Kevin Baas^{talk: new} 21:47, 2005 May 10 (UTC)

OK. Let's strip it down to this: Can we have chaos with a system whose representation in the phase space has an attractor of integral Hausdorf aka "fractal" dimension? I'm asking this mainly because it seems to me that the attractor proposed by XaosBits would be of integral H-dimension. Hence it wouldn't "produce information" and wouldn't be of "infinite complexity". Then was this a legitimate example of chaotic attractor or not?

I don't think "infinite complexity" has any real mathematical meaning here. And I don't know if integer hausdorf dimension neccessarily implies no information production. information production is equivocable to a positive lyaponuv exponent wich is equivalent to "divergence" - two neighboring trajectories diverge exponentially - notice no mention of fractals or dimensions at all, for that matter there. But I haven't seen any example of a integer-dimensional strange attractor, notwithstanding xaos' example, which i'm not convinced on, as I don't see any indication of or allusion to a positive lyaponuv exponent. 64.179.125.66 15:23, 11 May 2005 (UTC) this was me. Kevin Baas^{talk: new} 18:40, 2005 May 11 (UTC)

• (side note: i've never heard of such intermitent chaos having non-fractal structure ([[Beno%EEt_Mandelbrot|Mandelbrot]] actually defines "intermittence" as a temporal fractal. It's one event that led to his discovery of fractals. He discovered it when he was working for AT&T, trying to figure out the source of noise on a copper wire. His discovery led Claude Shannon to create information theory. (since mandlebrot showed that you can't get rid of the noise, shannon had to figure out a way to deal with the noise) )) Kevin Baas^{talk: new} 17:26, 2005 May 10 (UTC)
  • I noticed "intermittence" is not mentioned in the fractal or the chaos article, and that there is no article for it. Kevin Baas^{talk: new} 17:51, 2005 May 10 (UTC)
  • I noticed the "henon attractor" is described as "strange", and this appears as an exception to "must be continuous-time". Like I said, I don't know about "officially" - i guess the definition is still too informal. :-( Kevin Baas^{talk: new} 17:51, 2005 May 10 (UTC)

I think that what is commonly described as Henon attractor is in fact the Henon map, which is IICR just a section of the actual attractor. The beast itself is embeded in 3D phase space and continuous as well. It is a curve enroled in a donut-like shape.Koolniczka 19:37, 10 May 2005 (UTC)

Merging

Isn't a chaotic attractor the same thing as a strange attractor, which already redirects here? It just seems a bit redundant. Nicholasink 00:37, 2 June 2006 (UTC)

Thanks for the edits. I feel that the chaotic attractor could be a redirect to the strange attractor section and some of the material incorporated into the attractor page, except maybe for the metions to Lasoda. His work seems only to use chaotic attractors as an analogy and not in any mathematical sense. XaosBits 01:00, 2 June 2006 (UTC)

Attractiveness and being an equilibrium

Hi all. I can only find in wikipedia attractiveness as an attribute of equilibria. But being attractive can be a property of any set, since a point is locally attractive for a system if there exists d>0 such that $x(t)\rightarrow x0$ as $t\rightarrow \infty$ for every x with x(0) far from x0 less than d. Remark that locally attractive points are not necessarily equilibria, and, even if they are, they are not necessarily asymptotically stable equilibria. I know an example, which is a little long to write down. Of course, attractor as defined in the

can you help me to clarify this to myself, first, and then to wikipedia? --achab 16:07, 30 January 2007 (UTC)

Cleanup request

I added the request for clean-up since most of the maths is not formatted very well and some of the sentences are a little awkward (I've changed one portion, but a more careful review seems appropriate). Thomas 18:15, 1 March 2007 (UTC)

I cleaned up the article as much as I could and removed the cleanup tag, though it is still difficult to read by a casual reader. -Pgan002 02:09, 22 June 2007 (UTC)

First time at the article and found myself stuck here:

That is, if a is a point in the phase space, so that the state of the system at a certain time, then f(0, a) = a...

I think there's a typo here, "so that the state of the system at a certain time" doesn't make sense when it's followed by "then f(0, a) = a" 138.37.33.159 (talk) 13:31, 18 November 2009 (UTC)

New Image

I've worked from this page to generate some sample code for OpenGL. I'll release it soon, but for now I propose we place an image I generated. Check it out! http://lostindev.com/files/Attractor.jpg Nint22 (talk) 01:44, 4 November 2009 (UTC)

The article Attractor is incomprehensible

The section Motivation is too abstract, and all the sections lack good graphic illustrations. I suggest that the article starts with two simple examples: one in which the phase space is easy to visualize, but where the attractor is trivial (a damped pendulum, for instance), another in which the attractor is strange, but where the phase space is difficult to visualize. I have the following proposal for the latter example. This example has the advantage, that it can be referred to later in the article, where it should be explained how it is possible in practice to find attractors with surprising shapes (namely by construction of Mandelbrot sets):

If $f(z)$ is a rational complex mapping from the plane into itself, and if a point z is iterated by $f(z)$, that is, if we form the sequence of iterations $z_{k+1}=f(z_{k}),k=0,1,2,\dots ,z_{0}=z$, then the most usual is that the sequence converges towards a fixed point or a finite cycle (an attracting cycle). If this is the case, then the set of points z whose sequence of iteration converges towards the same fixed point or cycle, is an open set. Such a set is called a Fatou domain (see Julia set). In the left picture below is $f(z)=z^{2}+c(c=-0.5655+0.157i)$, and there are two Fatou domains: in the outer the sequences converge to ∞, in the inner the sequences converge to the left of the two points in the centre. Now we introduce a disturbance in the function so that is it no more complex differentiable (but still real differentiable): we replace it by $f(z)=z^{2}+x/2+c(z=x+iy)$. Then the iterations in the outer Fatou domain are still towards ∞, and we can colour as before, but in the inner Fatou domain the terminus for the sequences of iteration is no more a fixed point or a finite cycle, but an infinite set, and it is impossible to colour by the method we have used, instead we colour the Fatou domain black and draw its attractor (which in this case seems to be a simple curve, but that is not so: it consists of a system of threads lying infinitely close).

• 
  Convergence towards a fixed point
• 
  Convergence towards an attractor

(Gertbuschmann (talk) 09:13, 2 May 2010 (UTC))

I completely disagree. Introducing a complex and abstract example at beginning of the article and focussing on strange attractors before simpler concepts have been explained can only make the article more difficult to understand. Also, suddenly introducing a discrete system when the rest of the article is about continuous systems will confuse the reader. Gandalf61 (talk) 16:48, 2 May 2010 (UTC)

And I'm confused by the use of the word 'large' in the Mathematical Definition section thus: For any open neighborhood N of A, there is a large positive constant T such that f(t,b) ∈ N for all real t > T. Perhaps this phrasing has acquired a technical meaning since I did the Maths Tripos in the 1950s, but for me the word does not add anything to the statement it is in. Brian Josephson (talk) 11:27, 6 August 2010 (UTC)

I have removed "large" from that definition - it wasn't necessary. Indeed, if b ∈ N then it may be possible to make T as small as we like. Gandalf61 (talk) 12:18, 6 August 2010 (UTC)

Numerical localization (visualization) of attractors

Because of many pictures of attracters in the article, I think it would be good to add something about attractors visualization:

From computation point of view, attractors can be naturally regarded as self-exciting attractors or hidden attractors. Self-exciting attractors can be localized numerically by standard computational procedure, in which after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable equilibrium, reaches an attractor and computes it (such examples are classical attractor in Van der Pol, Beluosov–Zhabotinsky, Lorenz, and many others dynamical systems). While basin of attraction of hidden attractors does not contain neighborhoods of equilibria, and therefore hidden attractor cannot be localized by standard computational procedure. --nk (talk) 15:26, 28 October 2011 (UTC)

Axes of the attractor images

It seems to me that someone who doesn't already know much about attractors is going to look at the images of attractors and find them to be just pretty pictures with no explanatory content. In some (all?) cases, I think we could add the following to the caption:

Each point in the graph shows on the horizontal axis the value of the dynamic variable at some point in time, and shows on the vertical axis the value of the variable at a time that is later by a fixed increment.

Comments, anyone? Duoduoduo (talk) 16:21, 9 December 2011 (UTC)

I'm not sure which image you're referring to or that the statement is true for any of them. The Lorenz attractor for example is defined in three dimensions and the image shows a the path of a point in space. In most case we are only interested in the qualitative behavior, so we don't really care what the coordinates are specifically.--RDBury (talk) 21:34, 10 December 2011 (UTC)

Attracting period 3 cycle and its immediate basin of attraction

Sorry for my careless wording above, which I believe applies to a lot of attractor graphs but maybe not to these. But again, to the uninitiated they all just look like pretty pictures. Maybe at some appropriate points, perhaps in the captions or perhaps in a new section of the text headed "Graphical interpretation", you could insert what you just said: "the image shows a the path of a point in space".

About twenty years ago I published four papers on chaos, so even though my memory is a little rusty and I'm not really an expert, I think that if I can't understand the images then most readers won't be able to. And honestly, I can't fully understand the one on the right here for instance. Maybe it should say something like "... 3-cycle of the evolution of a point in two dimensions and its immediate basin of attraction: the three darkest points are the three cycle, the relatively dark areas are ..., and the gray areas are ...; points in the light blue area do not [or, do not all] converge to the three cycle."

Weakly attracting fixed point for complex quadratic polynomial

And in the one on the left, what are the dynamics? "Complex quadratic polynomial" doesn't answer this for me. And what are the three axes? Only after I enlarge the image twice and then peer really closely at it, I can see that one is the real axis, and one is the imaginary axis; since most reader won't blow it up, they won't see the real and imaginary labels. And what is the third axis? If the polynomial maps from the complex plane to the complex plane, why aren't there four dimensions -- is this a three-dimensional cross section of R⁴? Or is the third axis a measure of the frequency with which the points in the complex plane are visited?

I think some explanation is in order for all the graphs, for the uninitiated and for the semi-initiated. Duoduoduo (talk) 22:26, 10 December 2011 (UTC)

The image on the left is explained in detail in its information page. Basically the horizontal plane is the complex plane and the vertial axis represents the number of iterations. The connection between this and the article text is a bit tenuous imo so I'm not entirely convinced it should be there. In fact several of the images seem to be there for eye candy with little value in explaining the text.--RDBury (talk) 08:57, 11 December 2011 (UTC)