Talk:Arnold's cat map

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Contents 1 Is there really a period? 2 Picture contribution 3 T-space 4 Recurrence time and initial conditions 5 Stronger separation betwen continuous and discrete map 6 Image should be moved

[edit] Is there really a period?

Are we sure that after a certain number of iterations the original image is restored (see Poincaré recurrence theorem)? The Poincaré recurrence theorem doesn't say that a set of states (i.e. points) terurn close to it's original form, it says that every point return close to itself but the returning time for any single point of a set is completely different.--Pokipsy76 21:26, 28 June 2006 (UTC)

If the individual points return in time t1, t2, t3, ... then the whole configuration returns in time t = t1 * t2 * t3 ...

It can be proven that the integer cat map returns in at most 3N discrete timesteps. JocK 12:46, 30 June 2006 (UTC)

But that product is infinity if you consider a set that contains an infinite number of points (as the image of the cat is).--Pokipsy76 12:55, 30 June 2006 (UTC)

In reality it isn't. However, based on your comments I have decided to limit to discussion to integer cat maps. The restoring of the image is trivial for such maps. JocK 13:37, 8 July 2006 (UTC)

Why the standard cat map on the torus shouldn't be covered by this article? If you want to limit the discussion to the integer cat map maybe you should rename the article.--Pokipsy76 14:38, 8 July 2006 (UTC)

You're right. I think the latest edits address this issue. JocK 18:22, 9 July 2006 (UTC)

But the "original" Arnold's cat map is that on the torus. The discrete map is a subsequent development of the origional idea. The original cat map on the torus should come first, and only after that we can introduce the further developments.--Pokipsy76 09:55, 10 July 2006 (UTC)

I think that is beside the point. The current formulation applies to continuous and discrete domains. If discrete, the domain is a toroidal grid. JocK 19:03, 9 November 2006 (UTC)

Discrete and continuous cat map need different formulations and we should first give the formulation of the continuous one because it is the mist important and most known in the dynamical systems theory. Moreover properties of the classical (continuous) cat map like Anosov diffeomorphism and mixing are not shared by the discrete one and are erroneously attributed to it in the article.--Pokipsy76 14:36, 15 November 2006 (UTC)

Can you substantiate your claims? Are you suggesting that the integer cat map has no clearly marked local directions of 'expansion' and 'contraction'? Also, virtually each article on the cat map illustrates the mapping using a discrete image of a cat (or otherwise). I can't see why the continuous map should 'come first'. Anyway, the current article covers both. having said that: feel free to incorporate your ideas. JocK 19:20, 15 November 2006 (UTC)

Anosov diffeomorphism is a diffeomorphism i.e. a particular map un a manifold with some regolarity properties like differentiability. The discrete cat map is not defined on a manifold and is not differentiable in any sense. Having been studying dynamical system since some years as a graduate student I have seen the continuos cat map a lot of times (every book introducing ergodicity and mixing cite it) but I never heard or read of the discrete cat map before. This is why I think the continuos cat map is the origina one and the most known.--Pokipsy76 07:29, 24 November 2006 (UTC)

[edit] Picture contribution

Nice contribution, Claudio! JocK 18:57, 9 November 2006 (UTC)

[edit] T-space

What is ? —Centrx→talk • 16:47, 7 April 2007 (UTC)

is the standard notation for the n-torus.--Pokipsy76 08:45, 18 April 2007 (UTC)

Okay. —Centrx→talk • 19:56, 24 April 2007 (UTC)

[edit] Recurrence time and initial conditions

Concerning the picture that illustrates Poincare recurrence, the short time recurrences observed are not typical of chaotic systems. Actually, they are observed only because the trajectories used are all periodic (strictly speaking there is no chaos in the simulations). If chaotic trajectories are used the period increases dramatically. To clarify this point and estimate the recurrence times we made the following web page in our group. I think a link to this web-page and/or the use of some of its material would be useful in the main article. Edugalt 13:03, 19 July 2007 (UTC)

Thanks, have added a link. Would be good to have some of the content in the article itself. However, don't think we can really classify the trajectories of the integer cat-map as non-chaotic because these have a recurrence time scaling with the linear dimension of the area being mapped. On the contrary, I would argue that chaos in integer 2D maps of size N x N typically manifests itself as a number of orbits and cycle times per orbit both scaling with N. JocK 08:54, 20 July 2007 (UTC)

[edit] Stronger separation betwen continuous and discrete map

I think we need to split the article in two section, one for the continuous map and one for the discrete one because the actual form seems a little bit confusing. For example it seems to say that the continuous map becomes the identity map after a finite number of steps, while this is true just for the discrete one.--Pokipsy76 07:45, 30 July 2007 (UTC)

Both the real and the integer cat-map exhibit Poincaré recurrence. The consequence of this is that integer cat maps yield the identity map after a finite number of steps, whilst real-valued cat-map after a finite lapse of time return to a state arbitrarily close to the initial state. I.e. a "noisy" identity map is obtained, the noise level of which can be made arbitrarily small by cycling through more and more "close recurrences". having said that, I agree that the discussions on the continuous and discrete cat map need a bit of a re-write to make them more consistent. If I find the time, I will try and make some edits to accomplish that. Regards, JocK 19:21, 31 July 2007 (UTC)

Mmmmmh.... I don't think that a "Noisy identity map" is obtained and I will explain why. Poincaré recurrence theorem says that for the (continuous) cat map we have this: if we fix ε then for every point x there exists a number N(x,ε) such that iterating the map on x for N(x,ε) times will give a new point that is close to x less than ε. But the number N(x,ε) depends on the point x! There is not a number N(ε) depending only on ε such that the iteration of the map for N(ε) times on *every* point x is close to x less than ε!!--Pokipsy76 14:43, 1 August 2007 (UTC)

I agree that a separation is needed. I think the more relevant case is the continuous version, what is not clear from the present version. I would say that one paragraph at the end, with reference to the picture, should be enough for the discrete case. Moreover, I don't know what the terms "mixing" and "chaotic" mean for the discrete case. Concerning the Poincare recurrences, I would say they are trivial in the discrete case since all initial conditions correspond to periodic orbits and the recurrence time is just the period. In the continuous time case the periodic orbits have zero measure and in this sense all initial conditions are chaotic. I would be extremely careful with the terms "noisy" and "random" since these terms seem to contradict the obvious deterministic nature of the map (Edugalt 13:03, 9 September 2007 (UTC)).

[edit] Image should be moved

The image shows a behaviour of the discrete map which is not shared by the continuous map, therefore I think it should be moved on the appropriate section. Objections?--Pokipsy76 (talk) 14:46, 2 June 2008 (UTC)

An image like this one would be more appropriate for the continuous map.--Pokipsy76 (talk) 14:51, 2 June 2008 (UTC)

Y Done--Pokipsy76 (talk) 22:25, 7 June 2008 (UTC)