The article is generally good, and reasonably comprehensive. However it fails to discuss the most important work justifying the overlap criteria, that of John Green. The overlap criterion shows great insight but is not rigorous and only approximately determines a chaos boarder. Green's method supplied these deficiencies. I suggest, after references (see[2,3]), the following be inserted: "The overlap criterion, while supplying important insight to the onset of chaos, is not rigorous. A more rigorous approach was supplied by John Greene [4,5], who identified the chaos boarder with the destabilization of higher order fixed points. This construction also very accurately predicted the chaos boarder" The following sentence would also be slightly modified to insert "also" between "can" and "be".
4. J.M.Greene, J.Math.Phys. 9, 760 (1968) 5. J.M.Greene, J.Math.Phys. 20, 1183 (1979)
Another suggested change is at the end of the example, where the last sentence should be expanded to read "..with a few percent accuracy, provided the overlapping islands are of nearly equal size. For unequal size overlapping islands the criterion becomes less accurate."
Finally my amusing comment on the heading "Chaotic stories" If the author means stories about chaos, then the heading should be "Chaos stories". However, if the author means stories that are themselves chaotic, then the heading is correct. Perhaps the author wants the heading to have both meanings.
Author : to Referee A
To Referee A: thanks for your useful remarks. Of course, the Greene approach is more exact if one knows what is the last critical invariant curve and if one finds numerically the corresponding fixed points. This however is not always so simple to implement if one needs to know an analytical dependence on system parameters.
By itself, it would be very useful to have an article on Greene method, with further developments by R.MacKay, at Scholarpedia
I agree with referee A (Lichtenberg) but feel that his 2/3 rule has proved easy to apply for a wide variety of maps, especially when there is a substantial gradient in P. The author could point out conditions for validity of the criterion. There are situations (e. g. circular polarized microwaves) where the islands are highly distorted and the Chirikov scenario is inappropriate. Such maps should be excluded, or better variables found. The stories are fascinating but the article needs some details.
Author : to Referee B
The article is updated according to the suggestions of Referee B.
I made some minor changes in the wording of the article. Please look them over.
Author : to Editors and Referees
your modifications are accepted