User talk:Ilya Voyager

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OTRS confirmation needed

An editor, Leningradartist, claims to be the copyright holder of a number of images he has uploaded on to the Commons, and says that he has communicated this to the OTRS (see tickets #2010032110008071 and #2010040210038813). I've been in touch with an OTRS volunteer, Joe Daly, but he says he cannot read the e-mails in question as they are in Russian.

I understand from Joe that you were the OTRS volunteer who issued these tickets. Can you have another look at these tickets and confirm if one or both of them cover the image File:Bazhenov-Vsevolod-In Singapore-strait-geo36bw.jpg? If so, please add the {{OTRS}} tag to the image description page. Thanks. Otherwise, I will nominate the image for deletion. — Cheers, JackLee ^–talk– 19:57, 26 April 2010 (UTC)

Yes, this ticket is valid. Uploader claim that he have a lot of written permissions (copyright transfer contracts) with the authors mentioned in his book devoted to Leningrad artistic school, and we accepted these claims after discussion with Cary Bass. I'll check with tags soon. Ilya Voyager (talk) 07:37, 28 April 2010 (UTC)

OK, great. I'll leave it to you to apply the {{OTRS}} tag to the image description page. — Cheers, JackLee ^–talk– 11:30, 28 April 2010 (UTC)

 Done for this image. Ilya Voyager (talk) 19:53, 9 May 2010 (UTC)

Stream plot regarding Centre Manifolds

Hi Ilya,

I was on the Center Manifold Wiki page, and I believe you contributed the image there: Saddle-node phase portrait with central manifold. I have various issues with this image that I hope you can clarify for me.

First off:

What stream function is that a plot of? I had a look at the source code and couldn't figure it out (it seemed at point as though you had the stream function being piece-wise defined for x?)

Second:

The tile describes the plot as a Saddle node with a Central Manifold. a) The x > 0 section is not a saddle node. b) Saddle nodes don't have Central Maniolds. By definition, they have 2 real eigenvalues, one positive one negative.

Finally:

Whilst the green highlighted manfold is certainly an unstable manifold, the Red manifold is not a central one, for x < 0 is flow straight to the origin, so it's a stable manifold.

If you could explain to me why the image doesn't seem to represent the title or the page it's on, would be greatful, as I would hate to wrongly remove an image.

Many Thanks, Brondahl 131.111.184.4 09:19, 9 May 2010 (UTC)

Hi Brondahl, and thanks for your interest :)

1) This is a phase portrait of the standard saddle-node vector field: ${\dot {x}}=x^{2},\ {\dot {y}}=y$. It can be easily integrated, the phase flow looks like: $x(t)=1/(1/x_{0}-t),y(t)=y_{0}\exp t$. So if you are interested in phase curves, you have from the first equation $1/x=1/x_{0}-t\Rightarrow t=1/x_{0}-1/x$ and therefore $y(x)=y_{0}\exp(1/x_{0}-1/x)$. This corresponds to the line "return z0.y*exp(1/z0.x-1/x)" in the source (see the definition of function yt).

2) The saddle-node singular point is a non-elementary singular point that have some (but not all) zero eigenvalues (1 positive and 1 negative eigenvalue corresponds to saddle points, not saddle-node ones). See e.g. p. 542 of Methods of qualitative theory in nonlinear dynamics, Part 2 by Leonid P. Shilnikov, Andrey L. Shilnikov, Dmitry V. Turaev, where you can find the figure very similar to the discussed one (module symmetries) for $\mu =0$.

3) The red curve on the figure is a central manifold (actually, one of the possible central manifolds), because it tangents to subspace of the linearization matrix which corresponds to zero eigenvalue (this subspace is $y=0$). It neither stable nor unstable manifold because it do not correspond to any nonzero eigenvalues.

So, I believe this figure is correct. Ilya Voyager (talk) 14:27, 9 May 2010 (UTC)

Ah yes, I was completely wrong. Not sure why I missed node vs point. Also, I intended my original post to have much more "please explain the diagram" and much less "prove to me that you're not wrong", re-reading it seems that I missed the mark there. All-in-all I must be much more tired than I thought.

Could you explain one more thing to me, please... I was under the impression that a central manifold represented a section of the flow which moved dramatically more slowly than the rest. ie. that particles would tend to the centre manifold in O(x,y) and then flow along the manifold in O(x^2,xy,y^2). That doesn't appear to be what is happening in this case.

Is my understanding just plain wrong or is it only true in special cases, or what?

Brondahl (talk) 19:14, 9 May 2010 (UTC)

Actually, the particle which belong to center manifold tends to the singular point (when t tends either to $+\infty$ or to $-\infty$) as $O(1/t)$ and particle attracts/repells to/from center manifold (in hyperbolic direction) with rate $O(e^{t})$ which is much faster for $t\to +\infty$. (You see, that ${\frac {\exp(t)}{1/t}}\to 0$ as $t\to -\infty$.) This can be easily seen from the formulas for the flow discussed above. (I also added this formulas to the description page of the figure.) So it seems that your impression is valid for this case if applied properly. Ilya Voyager (talk) 19:50, 9 May 2010 (UTC)

Template:Asymptote

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