Talk:Blue-sky catastrophe

To the Reviewer: Thank you very much! Andrey

Andrey,

My comments are the following:

1. The authors define a blue sky catastrophe as a situation in which both the length and period of a stable limit cycle tend to infinity while the orbits remain in a bounded part of state space. From reading this article, it is not clear to me if there is (up to some topological equivalence) only one blue sky catastrophe. Has this been proved or what is the status?

We describe two other configurations for the catastrophe. Unfortunately, they are not of general position, i.e. not of co-dimension one. So far, the given configuration by L. Shilnikov and Turaev is the only one that meets the conditions.

2. In the case of Group I, I would prefer to see T ~ 2\Pi / \omega.

Done

3. The references should be prepared more carefully. For example:

- Is the second reference a paper or a book? Which journal or which editor?

The book, now specified.

- Paper of R.H. Abraham: is it really "chaostrophes"?

Oohh.... done

- The reference to the paper of V.S. Medvedev contains a MR number; other references do not include this.

Removed

- Paper of W. Li and others: presumably this is a Special Year?

Special indeed..

- Paper of N. Gavrilov and A. Shilnikov: remove the TeX style {\rm ibid}.

"ibid" as it published in the same place as the above reference.

Minor remarks:

1. In the caption of Figure 2 "magniture" should be "magnitude".

Done

2. In one place L. Shilnikov is written with two l's.

Done

3. In the caption of Figure 11: delete one "near".

Done 4. In the description of the leech heart interneuron model: replace i.g. by i.e.

Done

5. Just before the References: replace "low" by "law".

Done

Origins of the idea

In "Foundations of Mechanics", Ralph Abraham and Jerrold E. Marsden, 2nd edition, Benjamin/Cummings 1978, there is a phase portrait labeled "blue-sky catastrophe" on page 566-567, and that picture credits Ruelle and Takens for coming up with the idea in 1971. The reference given (I have not checked it) is

"On the nature of turbulence", Comm. Math. Phys. 20 pp 167-192 and 23 pp 343-344.

While the idea is clearly not as fully developed as here, it does state the basic idea: "As μ approaches zero from the left, the period of the attracting closed orbit tends to infinity." The (rather simple) picture implies that the measure evaporates to zero.

The period of a periodic orbit becoming a homoclinic one, i.e. bi-asymptotic to an equilibrium state tends to infinity. This makes the bifurcation truly homoclinic. The challenge of the blue sky catastrophe is to find configurations, other then "plain" homoclinic ones, without equilibria involved, that give the desired result. [reply by the article authors]

Yes, fine, and that indeed seems to be the modern contribution. I was merely trying to point out that the words "blue sky catastrophe" have been in use for a while, before your strong examples were discovered. Linas 11:21, 25 March 2008 (EDT)

P.S. The correct way to respond to a posting is to indent (with a colon); this makes the response easier to read, and to sign your reply with four tildes in a row, this: ~~~~. Linas 11:21, 25 March 2008 (EDT)

It also gives two "modern refs" as

J.C. Alexander and J.A. Yorke 1978 "Global Hopf Bifurcation", Am. J. Math, "to appear" (I guess, not yet in print when the book came out)

RL Devaney 1978 "Blue sky catastrophes in reversible and Hamiltonian systems" (preprint) but google knows all: Indiana University Mathematics Journal 26 (1977), 247-263.

Linas 02:25, 11 March 2008 (EDT)