Talk:Nontwist maps

This is a good review of Nontwist maps, but needs some revisions. First, there are many too many references! Scholarpedia guidelines specify keeping them to a MINIMUM. Second, the material on Tokamaks is inappropriately detailed; there are many textbooks. I corrected a few obvious typos,

Allow me to quote:

   "Keep references to the minimum, citing only major books and review articles. At the end of the article put "Suggested Reading" list pointing to recent books."  The fact that this requirement has been violated is no reason to repeat the injury to scholarpedia standards.

Please reduce the number of references. You can write a full review article for another journal.

There is no discussion of vortex pairs, one of the unique indicators of a nontwist system. This can be done qualitatively, as in my own review article in the Encyclopedia of Nonlinear Science.

The article gives the impression that nontwist is limited to 2D systems. Not so. 4D systems have been discussed rather nicely by Dullin and Meiss, In fact I recently learned of earlier work by Laskar for accelerators where the frequency map is shown to have a fold. One of the aims of Scholarpedia is to provide up to date information on current research, and 4D systems is one of these.

If you can find a good reference for tokamaks, please give it, but this is not an essential point. But you could use the space for 4D systems. Or for vortex pairs.

Author Wurm :

We thank reviewer C for the corrections of the typos and the additional reference, which we have included.

The reviewer is correct in stating that the Scholarpedia guidelines suggest minimizing references. During the writing of the article, we were well aware of this guideline, but we observed that it is not strictly enforced (e.g. the "Chirikov standard map" section has numerous references) and felt it necessary to include references for the following reasons:

1. There has been a great deal of rediscovery in the area of nontwist maps, with different terminology, the same effects going by different names, etc. Many researchers appear to be unaware of all the work that has been done. So, one of our goals with this article was to make this clear and to identify and give credit to the references that made original and important contributions.

2. Nontwist maps, like the standard map, arise in many different physical contexts. We aimed to show this wide range of applications. For this reason, we tried to list the first occurrence of nontwist maps in each field of application. (incidentally, for this reason we have included and are especially appreciative of the additional reference (of which we were not previously aware) that the referee has provided). Although the list of papers is long, it is far from exhaustive. Many papers have been published based on the original works in the different fields.

3. The area of nontwist maps is still relatively young and the main results cannot be found in a book or in a review article. In most cases, the only reference is to the original research paper, so anybody reading the article who is interested in pursuing this topic further will need to start with these papers.

Regarding, the tokamak description, we tried to give the bare minimum of information so that somebody who does not know anything about nuclear fusion would get a sense of what this application was about. If reviewer C feels strongly about shortening the description, we can do that and add a general tokamak reference.

===============Reviewer A========================

I suggest some revisions.

1) Instead of the text:

"Apart from their physical importance, nontwist maps are of mathematical interest because standard proofs of important theorems concerning area-preserving maps assume the twist condition, e.g., the Poincare-Birkhoff theorem, the KAM theorem, and Aubry-Mather theory;"

that sounds incomplete, I suggest the following one:

"Apart from their physical importance, nontwist maps are of mathematical

interest because they do not satisfy the basic assumptions of the celebrated 
Poincar\'{e}-Birkhoff theorem, the KAM theorem, and Aubry-Mather theory,
and thus new results concerning their typical behaviour have to be derived;"

2) The definition of SNM is incomplete: it is not explained what \( \mathbb{T}\) means, and on the line of definition of \( x_{i+1}\), (modulo 1) is missing.

I suggest to define the map, not through a recurrence relation, but specifying its classical definition:

It is defined by 

<math >(x,y)\in \mathbb{T}\times\mathbb{R}\to(\overline{x},\overline{y})\in\mathbb{T}\times\mathbb{R},</math> where

<math stnt> \begin{array}{lcl} \overline{x} & = & x + a \left( 1-\overline{y}^2\right),\quad ({\rm modulo} 1)\\ \overline{y} & = & y -b \sin\left(2\pi x\right) \end{array} </math>

3) In the definition of a nontwist map, I suggest to replace the twist condition, with the nontwist one. In my opinion, the reader could understand better what the nontwist means from this condition: <math nontwist> \frac{\partial \overline{x}\left(x,y\right)}{\partial y}=0 </math>

3) I suggest to include in the text the remark that you refer only to rotational periodic orbits and invariant circles.

4) An invariant circle of both twist and nontwist maps contains more than one "irrational orbit", hence the text below:

"One of the main characteristics of nontwist maps is that more than one irrational orbit (and more than one chain of periodic orbits of the same winding number) can exist."

must be changed. I suggest to include instead, the following lines:

"One of the main characteristics of nontwist maps, in comparison to twist maps, is that more than one

rotational invariant circle, and more than one rotational chain

of periodic orbits, of the same winding number, can exist".

7) The same objection, relative to the text:

"When two quasiperiodic orbits collide, the winding number profile shows a local extremum (Fig.<ref>F2</ref>) and the orbit at collision is referred to as the shearless curve or shearless invariant torus."

I suggest instead, the following lines:

"Computer experiments revealed that when two rotational invariant circles of the same rotation number collide, the winding number profile exhibits a local extremum point (Fig.<ref>F2</ref>), and the resulting invariant set at collision is referred to as the shearless invariant torus or twistless circle ( Dullin et al., 2000)."

For a more theoretical definition of the twistless circle, see Dullin's paper.

8) In the text below:

"Under suitable changes of parameter values,  periodic orbits of the

same winding number approach each other in coordinate space."

I suggest to insert "phase space", instead of "coordinate space".

9) The phrase "Prior to collision, the orbits interact through chaotic layer reconnection ("separatrix" reconnection)" can be misleading, inducing the idea that rotational periodic orbits of the same period, always can reach the collision threshold, while for some regions in the parameter space there exist twin chains of odd periodic orbits that reconnect, but they never collide (as you revealed in your papers).

10) "The hyperbolic manifolds of the orbits connect" is inappropriate. In differential geometry, a

hyperbolic manifold is a special Riemannian manifold, and at a moment Scholarpedia
will provide a link to that notion. I suggest to say: "At a threshold the invariant
manifolds of distinct hyperbolic  periodic orbits, of the same period, intersect each other".

Author Wurm :

We thank Reviewer A for her detailed comments, believe her suggests improve the article, and have implemented all of them except for a slight wording change for #1 and #7 (reviewer A's numbering) and

Number 3 (1st) We feel that the suggested definition is not adequate. To be generically nontwist, there has to be a traversal of zero. The suggested definition allows for a tangency, which is violates the twist condition but is not generic.

Number 3(2nd) Instead of adding a sentence we added the word "rotational" a few times in the text to indicate that these are the only orbits we are talking about.

==============Reviewer A=============================

1) For: "Generically, nontwist regions have been shown to appear in area-preserving maps that have a tripling bifurcation of an elliptic fixed point.(van der Wheele et al., 1988; Moeckel, 1990; Dullin et al., 2000)"

I suggest:

"Generically, nontwist regions have been shown to appear in a neighbourhood of an elliptic fixed point of an area preserving map, undergoing a tripling bifurcation (van der Wheele et al., 1988; Moeckel, 1990; Dullin et al., 2000)"

2) The text "where \(c\) is a positive real number," after the twist condition can be removed.

3) I think that it is sufficient to say "A "nontwist" map violates the twist condition locally in a generic way, with a simple traversal of zero.

4) The line "One consequence of the violation of the twist condition is the existence of multiple periodic orbits with the same winding number" is not quite appropriate, because even twist maps can have two distinct periodic orbits of the same period, one of negative residue and the second of pozitive residue.

5) The following phrase "A nonexhaustive selection of references is provide here to provide an entry-way into the literature".

repeats "provide". I propose: "A nonexhaustive selection of references is given here to provide an entry-way into the literature."

6) I suggest to introduce a reference to the rotation number definition, inside the text:

"One of the main characteristics of nontwist maps, in comparison to twist maps, is that more than one rotational invariant circle, and more than one rotational chain of periodic orbits of the same winding number (rotation number) can exist".

7) I suggest to replace the lines: "In the standard nontwist map of Eq.(<ref>stnt</ref>) this can be seen easily in the integrable case where \(b=0\): two orbits with winding number \(\omega\) are located at

<math integrable-po>

(x,y) = \left( x,\pm\sqrt{1-\omega/a}\right) \,,
</math>

i.e., two such orbits are found on any vertical line in phase space. For ,\(b \neq 0\), even more than two rotational orbits with the same winding number may exist."

with

"In the integrable standard nontwist map (the map Eq.(<ref>stnt</ref>) corresponding to b=0), for example, the invariant circles \( y=y_0 \), \( y=-y_0 \), \( y_0\in\mathbb{R} \), have the same rotation number \( \rho(y_0)=a(1-y_0^2)\)."

8) I suggest to replace the remark: "(In the integrable case, the point of collision occurs at \(a=\omega\))" with the following one:

"(In the integrable case, two invariant circles of rotation number \(\rho<a\) approach each other as \( a \) decreases and collide at \( a=\rho \))"

9) The caption of the figure 1 "Example of the (x,y)-space for standard nontwist map at at a = 0.615,b = 0.4."

is, in my opinion, more appropriate in the form:

"The phase portrait of the standard nontwist map corresponding to a = 0.615, b = 0.4."

==============Reviewer A=======================

I made a minor modification: within "in the neighbourhood" I inserted "a": "in a neighbourhood".