Talk:Noninvertible maps

Reports on the Scholarpedia article "Noninvertible Maps"

By C. Mira

Reviewer A

The topic of discrete dynamical systems obtained by the repeated application (or iteration) of maps has become a central and interdisciplinary research field, as models expressed in the form of discrete dynamical systems are now very common in physics, biology, economics, social sciences. The phenomena related to the complex dynamic behaviours observed in these models is now well known even among non specialists. When the iterated maps are noninvertible the understanding of the global dynamic properties becomes particularly challenging, as their geometric properties give rise to particular topological structures of the attractors and basins of attraction. These geometric properties that characterize the behaviour of iterated noninvertible maps are well described in this paper, written by one of the pioneers in this field. Starting from one-dimensional maps the most important concepts are defined and extended to two-dimensional ones with the proper choice of figures and examples. The style of exposition emphasizes one of the main feature of this research topic: the interplay among analytic, geometric and numerical methods. I only suggest a few minor corrections:

In the section “Singularities common...” the sentence: An hyperbolic cycle is attracting if, and only if, all the multipliers are such that |Sj|<1

should be changed into

An hyperbolic cycle is attracting if all the multipliers are such that...

In the same section the sentence “is not an integer (p-1)<d<p” should be changed into “is not an integer d<p”

Reviewer B

The paper deals with a very important research topic, and the author is one of the leading experts in this field. Since the discovery of complex phenomena in the one-dimensional Myrberg’s map this research subject had an increasing influence in the field of discrete dynamical systems and its many applications, leading to the understanding of complex dynamic behaviors in engineering, mechanical and physical problems, but also in several social sciences. The paper is well written and clear, briefly summarizing the important aspects understood and developed up to now. Only a few minor corrections are suggested here below, mainly of style: Pag.2 change “real root” into “real roots” Pag.4 change “transcendental” into “trascendental” Pag.4 change “if, and only if, all” into “if all” Pag.5 change “stable manifold (or stable set)” into “stable manifold” Pag.5 change “unstable manifold (or unstable set)” into “unstable manifold” Pag.5 change “is not an integer (p-1)<d<p” into “is not an integer d<p” Pag.6 change “Most of results” into “Most of the results”

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Reviewer C

Review of C Mira article Noninvertible Maps for Scholarpedia June 8, 2007

This review has three sections: I.General Comments, II. Reorganization Suggestions, and III. Editing Suggestions.

I. General Comments

This article contains many interesting examples of behavior appearing in noninvertible maps of the plane. The author is certainly one of the world's experts in 2D noninvertible phenomena, and the examples he presents are all relevant and interesting. The emphasis on two-dimensional examples is an appropriate choice. The extensive literature on one-dimensional maps is appropriately left to other Scholarpedia articles, such as the logistic map, and quadratic maps. The emphasis on noninvertibility due to the folding of phase space along critical curves for smooth maps is also an appropriate choice since this is the simplest type of 2D noninvertibility.

Some reordering of the topics presented, however, would make the differences between invertible phenomena and noninvertible phenomena clearer, especially to the novice. In addition, some topics should be included or emphasized more than they currently are. Briefly, the critical sets (usually curves in 2D) should be introduced first. Most of the phenomena unique to noninvertible maps are due to the interaction of the critical curves with various invariant sets: fixed and periodic points, stable and unstable invariant manifolds (more generally invariant `sets'), attractors, including invariant circles, basins of attraction, and basin boundaries. Examples of noninvertible phenomena for individual maps should be presented first, followed by bifurcations leading to these examples as parameters are varied. Codimension-one bifurcations should be sufficient for this article. The examples presented are excellent, but it would be more instructive to point out which bifurcations are unique to noninvertible maps.

II. Reorganization Suggestions

Change the title of the article to something like "2D Noninvertible Maps". A suggested article outline follows. The footnotes, in parentheses, are to provide more information on areas that are not simply reordering of topics already included in the current article. References, in square brackets, are either already in the author's reference list, or they are given below.

Outline:

• Introduction (1)
• Some definitions and notation
  • Critical curves

• Interaction of critical curves with various invariant sets
  • fixed points or periodic orbits with zero eigenvalue (2)
  • self intersections of invariant sets (due to folding)
    • (Un)stable sets vs manifolds
  • disconnected stable manifolds (due to multiple preimages) [EKO]
  • chaotic attractors (stretching and folding with the folding due to noninvertibility) (3)
  • absorbing and chaotic areas
  • disconnected and multiply connected basins of attraction
  • fractal basin boundaries (if different from noninvertible case)

• Bifurcations arising from interactions of critical sets with various invariant sets
  • fixed points or periodic orbits changing from locally orientation preserving to orientation reversing
  • creation of self intersections on invariant curves
    • via tangency [EKO, FGKMM, KOP]. Contact bifurcations?
    • via smooth to cusp to loop bifurcation [FGKMM, KOP]
  • creation of disconnected component of stable manifold [EKO]
  • noninvertible routes to chaos, including smooth invariant circle to weakly chaotic rings, to full noninvertible chaos (3)
  • bifurcation from noninvertible-type basin of attraction to absorbing area
  • bifurcation from simply to multiply connected basins (4)
  • bifurcation from smooth to fractal basin boundaries

• More complicated noninvertible phenomena (5)
• Analogues for higher-dimensional spaces (brief comments only, CM vs LC, for example)
• References
• See also

Footnotes corresponding to above numbers.

(1) Replace with something more like the author's paper [FGKMM] that is more specifically two-dimensional. Introduce the notion of critical curve \(LC_{-1}\) (and \(J_0\)), and the foliation of the plane according to the number of preimages which exist. The main difference between invertible and noninvertible maps is the interaction of invariant sets (see the list given in the second paragraph of the General Comments section above) with \(LC_{-1}\).
(2) The noninvertible version of the stable manifold theorem applies ([R], [MS]).
(3) See Lorenz [L]. There is a sequence of four figures illustrating a route to chaos via the breakup of an invariant circle in Fig. 3. Fig 3d shows a typical picture of a chaotic attractor (and an absorbing region).
(4) The author's example is a great one. See also [AKL] and [EKO].
(5) Optional section: mention less basic examples such as those with vanishing denominator, examples of transverse homoclinic points without chaos [S], or codimension-two bifurcations [KOP].

[FGKMM] Frouzakis, et. al. (1997) (Already in the author's list of references)

[AKL] Adomaitis, R. A., I. G. Kevrekidis, and R. de la Llave (2007), A Computer-Assisted Study of Global Dynamic Transitions for a Non-Invertible System, Int. J. Bifurcation Chaos 17, 1305-1321.
[EKO] England, Krauskopf and Osinga (2005), Bifurcations of stable sets in noninvertible planar maps, Int. J. Bifurcation & Chaos 15(3): 891-904.
[KOP] Krauskopf, Osinga and Peckham (2007), Unfolding the cusp-cusp bifurcation of planar endomorphisms, SIAM J Appl Dynamical Systems 6(2), 405-440.
[L] Lorenz E (1989) Computational Chaos -- A prelude to Computational Instability, Physica D (35), 299-317.
[MS] McGehee and Sander, A new proof of the stable manifold theorem, ZAMP, 1996, 497-513.
[R] Robinson C, Dynamical Systems, CRC Press, Boca Raton, 2005.
[S] Sander E (2000) Homoclinic tangles for noninvertible maps, Nonlinear Analysis (41) 259-276.

III. Editing Suggestions (Some of these may become irrelevant upon revision.)

A. Notation:

• 1. Both x and X are used as a point in the space. It seems to be more standard notation to use X as the space (F:X->X) and x for a point in the space. The notation chosen should be consistent throughout the article. Then, for example, dim (X) =1 refers more naturally to the dimension of the space. X should be at least a smooth manifold so that, for example, eigenvalues can be defined.
• 2. T and F are used for names of the map. It is said in the first section that F defines T, but T is not actually defined. It would be clearer to just stick with one letter or the other.
• 3. In Figs 1, and 2, it appears that x(C) means the x-value of C, but this notation is not defined. The notation C would be clearer than x(C).
• 4. Sets are defined, for example, by x<C. It would be clearer to use {x:x<C}, or at least {x<C}.
• 5. `X=T<sup>-1</sup>' is used three times just before {X: X=TX}. As the author states, T<sup>-1</sup>x is a set, not a point, so having `X=' is misleading. It seems better to first define T^-1y={x: y=Tx}, then point out that this set could be empty (as opposed to not existing), or a single point, or many points, according to the number of preimages of x.
• 6. Introduce the alternative notation \(J_0\) as an (almost) synonym for \(LC_{-1}\). Give the French origin of LC. Comment on the differences between \(LC_{-1}\) and \(J_0\). Same for \(J_k\) as (almost) a synonym for \(LC_{k-1}\). Leave the more general CM for comments at the end for higher-dimensional analogues. Make this section more like the author's 1997 IJBC paper [FGKMM].

B. Other editing comments in the order they appear in the article:

• Introduction section:
  
  • 1. Fig 2: change `extrema of f(x)' to `extrema of f'
    
  • 2. Change 'Thus a two-dimensional quadratic map can belong either to' to 'For example, a two-dimensional quadratic map can either be invertible (the Henon family), or belong to'
    
  • 3. The comment about difference equations could be toned down, for example to `Compare recurrences with difference equations ... '

• Definitions section:
  
  • 1. Change `the T is called' to `then T is called'
  • 2. The reference to Myrberg's maps should also mention that this map is now commonly referred to as the quadratic map, and is essentially equivalent to the logistic map, and provide references to quadratic maps and logistic map (not written as of this date)

• Singularities Common ... section.
  • 1. The term singularities is not defined. Changing singularities to invariant sets seems more appropriate and more precise. In addition, first time readers might confuse this with the critical set where the Jacobian matrix is singular. The terms in this section are (or should be) already defined in other Scholarpedia articles: fixed points, periodic points, stable/unstable manifolds. and invariant circles. Attractors are mentioned, but attracting sets are not. Attractors and attracting sets should be defined before basin of attraction and then basin boundary, although just listing these invariant sets (with links) and saving more detail for the phenomena specific to noninvertible sets seems appropriate. Existing Scholarpedia articles by Milnor (attractors) and Ott (basin of attraction) already exist and provide good definitions of these terms. (Note: The term repelling is usually used as a synonym for expanding. Saddles are unstable, but not repelling.)

• Section 3.1 Definitions and some properties.
  • 1. It is not clear what `Exceptionally non critical lines in the above sense' means
  • 2. A cycle with +1 and -1 multipliers is not unique to noninvertible maps. It occurs in two-dimensional orientation-reversing maps. Examples of codimension-two bifurcations unique to noninvertible maps are given in [S] and [KOP].

• Figure 3 and accompanying text in Section 3.2: Basins and their bifurcations:
  • 1. It is not clear which `strict inclusion' is being referred to.
  • 2. The dynamics of the map would be clearer if the Attractor A were either included and labeled, or at least described.
  • 3. Change `turn into a points set' to `turn into a set of points'.

• Section 4: Absorbing and chaotic areas
  • 1. Provide precise definitions of absorbing area and chaotic area.

• First subpage:
  • 1. Example is misspelled
  • 2. It would be instructive to know which bifurcations are unique to noninvertible maps.

Reply to review C

Preliminary answer

First and foremost I would like to thank warmly the reviewer for his long, careful and precise analysis of the entry, which leads to improve the article. I am aware of the amount of work, and time devoted to this task, surely in spite of heavy academic responsibilities.

Before answering to more particular points, I would like to discuss the main point of the review, i.e. the reorganization of the article. The first point of my comment is about the fact that the topic of the article is not limited to two-dimensional maps, contrarily to what the review implies, and its proposition of a full reorganization states clearly. Thus the aim of the article is reminded: it is clearly displayed as dealing with p-dimensional noninvertible maps, with examples for p=1,2, using the continuity hypothesis, the restriction to smoothness being indicated when this is necessary.

Noting the necessity of including and emphasizing some topics not presented, or not sufficiently developed in the present entry, the review C proposes a full reorganization of the article, which surely would lead to an interesting long paper in a journal. The difficulty is that the text was written as an encyclopedia article (characteristic of Scholarpedia), so submitted to length limitations, which constitutes a technical obstacle to put into practice this suggestion. In one of his mails the Editor-in-Chief reminded me this point.

Even limited to p-dimensional, p=1,2, noninvertible maps, the topic is a vast, far-reaching one which would require several articles in Scholarpedia. The present entry is only a general skimming over the field, at the extreme length limitations asked by Scholarpedia. It is the reason why, in particular, the article does not develop some interesting and important situations. Among them, for example the ones created by maps without denominator, but with an inverse having a vanishing denominator, this concerning an important class of maps. So quoting briefly this case about its consequences on some basic points of the article, it gives a link with the entry "Maps with vanishing denominators", which is a way to provide a more developed information.

I fully agree with the importance of the phenomena, unique to noninvertible maps, due to the interaction of the critical curves with various invariant sets. This especially as I appear as co-author in the Frouzakis CE., Gardini L., Kevrekidis I. G, Millerioux G. & Mira C. [1997] paper, quoted by the review, and that I refer to. Its manuscript (discussed long before the publication with I. Kevrekidis) laid the foundations of the chapter 6 of the Mira et al. . [1996] book, and has been the "germinal" publication from which several other important papers (quoted in the review) were written. Also I am all the more aware of the importance of these phenomena, because I described such phenomena in the old papers (the oldest in 1969) quoted and presented in the two 1980 books (with I. Gumowski), related to interaction of a critical curve with a basin boundary, and with the unstable manifold of a saddle point, for p=2.

I go further: one-dimensional noninvertible maps also generate the interaction of an invariant set (a basin boundary, invariant for the inverse map) with a critical set (a point). In the past I developed results in this domain, and I would have liked not to be limited by length limitations to present them in the entry. So a chapter is devoted to this question in my 1987 book (World Scientific) (see also the 1980 book in French), and in the Mira et al. [1996] book (also see the older references of these three books). Due to Scholarpedia rules I chose to give examples for p=2 after the Figs. 1 and 2. Nevertheless, in view of complementary entries about the one-dimensional case , other authors can find the basic information in the above quotations. They can also find data about the interesting problem of imbedding of a one-dimensional noninvertible map into a two-dimensional invertible one, and the fractal "box-within-a-box" bifurcation structure generated by quadratic maps I introduced from 1974 (about the quotation of this structure cf. Guckenheimer J., 1980, "The bifurcations of quadratic functions", N.Y. Acad. of Sci., 75/1, p. 343-347). All these topics are related to the interaction of a critical point with the stable set of an unstable fixed point (the point and its increasing rank preimages.

Having said all these points about the limited aspect of the present entry, I suggest an alternative proposal about the introduction of a more developed presentation of the interaction of critical curves with various invariant sets for two-dimensional noninvertible maps. Indeed Yannis Kevrekidis is the leader for most papers quoted in review C. I propose that he becomes the author of an entry, whose title might be "Two-dimensional noninvertible maps. Interaction "critical curve-invariant sets", or simply "Two-dimensional noninvertible maps". This solution would allow me to create links with this possible new entry, giving to a reader complementary information.

I am aware of the fact that the article is not perfect, because resulting from a compromise between opposite constraints: development of interesting points and length limitations. I tried to optimize this problem, which was accepted by reviewers A and B.

Answer to the different points of review C

I. General Comments This article contains many interesting examples of behavior appearing in noninvertible maps of the plane. The author is certainly one of the world's experts in 2D noninvertible phenomena, and the examples he presents are all relevant and interesting. The emphasis on two-dimensional examples is an appropriate choice. The extensive literature on one-dimensional maps is appropriately left to other Scholarpedia articles, such as the logistic map, and quadratic maps. The emphasis on noninvertibility due to the folding of phase space along critical curves for smooth maps is also an appropriate choice since this is the simplest type of 2D noninvertibility.

Answer

I would like to make a remark about the extensive literature on one-dimensional maps, and on more general maps, noting that from the years 1975, when the maps topic progressively has become popular in the English literature, but very rare cases, it has been developed ignoring the basic publications made in other languages. These publications are numerous.

- The French school of iteration of the end of the 19 th century and the beginning of the 20 th one, with in particular the Lattes contribution.

- The Italian school with Levi Civita and Cigala (I used his results for constructing normal forms in 1970-1971).

- The Russian Andronov' school from the years 1935, and after 1950 with Pulkin, Neimark, Leonov (whose results on piece-wise continuous one-dimensional maps were rediscovered in many papers), nevertheless Shilnikov and his co-authors escaped this silence, ditto for Sharkoskij (Ukrainian school).

- More related to one-dimensional maps: the Myrberg's fundamental papers (1958-1963, in German, one in French), on the basis of which I developed the identification of the fractal "box-within-a-box" bifurcation structure (from 1974) generated by quadratic maps, using the interaction of critical points with unstable cycle points (cf. details in my 1987 book). So the Myrberg's papers define the cascade of period doubling period (called "spectrum") and their accumulation value (for the doubling from the fixed point with negative multiplier, approximate value given by Myrberg: 1.401155189). They give the cycles ordering laws from a binary code, also what is now called "Misiurewicz points". Implicitly they contain the popular notions of invariant coordinate, kneading invariant. All these results were after attributed to other authors, their topic presentations becoming undeservedly the standards. This justifies my quotation of Myrberg alone (see below).

About this topic I wrote several historical texts: chapter 1 of the 1980 French book, preface of the 1987 book, and in:

- Introductory presentation of International Colloquium "Point Mapping and Applications" (Toulouse, Sept 10-14, 1973) Proceedings Editions du CNRS Paris, 1976,

- Mira C. [1997] "Some historical aspects of nonlinear dynamics. Possible trends for the future". Double publication:(1) Intern. Journal of Bifurcation and Chaos, vol. 7, n° 9&10, 1997, 2145-2174. (2) The Journal of the Franklin Institute, vol. 334B, n° 5/6, 1997, 1075-1113.

About the anteriority of papers not written in English also see:

A. Dahan & D. Aubin, “ Writing the History of Dynamical Systems and Chaos " Historia Mathematica 29 (2002) 1-67.

Review part

Some reordering of the topics presented, however, would make the differences between invertible phenomena and noninvertible phenomena clearer, especially to the novice. In addition, some topics should be included or emphasized more than they currently are. Briefly, the critical sets (usually curves in 2D) should be introduced first. Most of the phenomena unique to noninvertible maps are due to the interaction of the critical curves with various invariant sets: fixed and periodic points, stable and unstable invariant manifolds (more generally invariant `sets'), attractors, including invariant circles, basins of attraction, and basin boundaries. Examples of noninvertible phenomena for individual maps should be presented first, followed by bifurcations leading to these examples as parameters are varied. Codimension-one bifurcations should be sufficient for this article. The examples presented are excellent, but it would be more instructive to point out which bifurcations are unique to noninvertible maps.

Answer

See section 1 of the reply.

Review part

From II. Reorganization Suggestions

.../... to "Footnotes corresponding to above numbers."

Answer

The sec. 1 of the reply answers to this part of the review. About the five footnotes:

Footnote 1. The purpose of the article is not the presentation of the two-dimensional case, but a short overview on new characteristics introduced by noninvertibility in the general case. Two-dimensional maps are considered as illustrative examples. In this framework, as said above the text was written as an encyclopedia entry (characteristic of Scholarpedia), thus submitted to its related limitations. So its topic, and its presentation are basically different from the outline of the paper [FGKMM]. As recommended by Scholarpedia the introduction must be the simplest possible. It is why first a basic property, related to the division of the space into open regions, is given as being the main difference from invertible maps. The interactions mentioned by the review C are only consequences of this property . For a pedagogical aim, the introduction illustrates this property from the one-dimensional case, in particular with Figs 1 and 2 introducing the simplest case of critical set, the critical point one.

As for the other footnotes the length limitations of a Scholarpedia article do not permit such developments. An entry by Yannis Kevrekidis (the leader for all the papers quoted by the review) with a title "Two dimensional noninvertible maps", would allow to give such an information via links that I will add. I know the references given by the review, but Scholarpedia recommends to limit the number of quotations. It is why I do not quote the "germinal" papers dealing with the "interaction" topic, such as:

- Mira C. & Roubellat J.C.[1969] "Cas où le domaine de stabilité d'un ensemble limite attractif d'une récurrence n'est pas simplement connexe". Comptes Rendus Acad. Sc. Paris,. Série A, 268, 1657-1660.

- Gumowski I. & Mira C., [1977], "Solutions chaotiques bornées d'une récurrence ou transformation ponctuelle du second ordre à inverse non unique", Comptes Rendus Acad. Sc. Paris, Série A, 285, série A, p. 477-480.

- Gumowski I. & Mira C., [1978], "Bifurcation déstabilisant une solution chaotique d'un endomorphisme du 2nd ordre", Comptes Rendus Acad. Sc. Paris, Série A,. 286, série A, p. 427-431.

And other previous publications quoted in the chapter 6 "Igor Gumowski and the Toulouse research group in the prehistoric times of chaotic dynamics" of the book "The chaos avant-garde. Memories of the early days of chaos theory" [2000] . World Scientific Series on Nonlinear Science. Series editor L.O. Chua. Series A, vol. 39, 219 pages.

Ditto I do not quote the paper:

Mira C., G. Millerioux, J.P. Carcasses, & L. Gardini [1996b]. Plane foliation of two-dimensional noninvertible maps. Int. J. of Bif. and Chaos, 6 (8), 1439-1462.

Review text

III. Editing Suggestions (Some of these may become irrelevant upon revision.)

A. Notation:

• 1. Both x and X are used as a point in the space. It seems to be more standard notation to use X as the space (F:X->X) and x for a point in the space. The notation chosen should be consistent throughout the article. Then, for example, dim (X) =1 refers more naturally to the dimension of the space. X should be at least a smooth manifold so that, for example, eigenvalues can be defined. See the modification: x is now used only for a point of the one-dimensional space.

Answer

See the modification: x is now used only for a point of the one-dimensional space.

Review text

• 2. T and F are used for names of the map. It is said in the first section that F defines T, but T is not actually defined. It would be clearer to just stick with one letter or the other. See the new presentation of the article introduction and its section 1 about this point.

Answer

See the new presentation of the article introduction and section 1 about this point.

Review text

• 3. In Figs 1, and 2, it appears that x(C) means the x-value of C, but this notation is not defined. The notation C would be clearer than x(C).

Answer

See the modification.

Review text

• 4. Sets are defined, for example, by x<C. It would be clearer to use {x:x<C}, or at least {x<C}.

Answer

See the new caption of Figs. 1 and 2.

Review text

• 5. `X=T-1' is used three times just before {X: X=TX}. As the author states, T-1x is a set, not a point, so having `X=' is misleading. It seems better to first define T-1y={x: y=Tx}, then point out that this set could be empty (as opposed to not existing), or a single point, or many points, according to the number of preimages of x.

Answer

See the corresponding modifications in sec. 1.

Review text

• 6. Introduce the alternative notation as an (almost) synonym for . Give the French origin of LC. Comment on the differences between and . Same for as (almost) a synonym for . Leave the more general CM for comments at the end for higher-dimensional analogues. Make this section more like the author's 1997 IJBC paper [FGKMM].

Answer

In the smoothness case, J_0 is an alternative way of denoting \(LC_{-1}\) related to the set of points for which the map jacobian is zero. Voluntarily the J_0 symbolism (used in [FGKMM), is not introduced in the article, because the map does not need to be smooth (cf. sec. 1). The locus of the merging rank-one preimages \(LC_{-1}\) is independent of the map smoothness, as said in section 3. So many examples of continuous piecewise linear maps are given in the 1980a,b and 1996 books.

Review text

B. Other editing comments in the order they appear in the article:

• Introduction section:

1. Fig 2: change `extrema of f(x)' to `extrema of f'

Answer

Correction made.

Review text

2. Change 'Thus a two-dimensional quadratic map can belong either to' to 'For example, a two-dimensional quadratic map can either be invertible (the Henon family), or belong to'

Answer

Correction made.

Review text

3. The comment about difference equations could be toned down, for example to `Compare recurrences with difference equations ... '

Answer

Unfortunately it is not exceptional to meet papers calling "difference equation" what is a "recurrence relationship". This is a relatively common error to be underlined, so without necessity to be toned down, this for a pedagogical aim

Review text

• Definitions section: 1. Change `the T is called' to `then T is called'

Answer

Correction made

Review text

2. The reference to Myrberg's maps should also mention that this map is now commonly referred to as the quadratic map, and is essentially equivalent to the logistic map, and provide references to quadratic maps and logistic map (not written as of this date)

Answer

After the Myrberg's quotation, now the text mentions that the equation is equivalent to the logistic one. Other references cannot be given due to the limitations of the Scholarpedia "For authors". About the introduction of these references see above the reply sec. 2 (first paragraph). The important is to give the source of fundamental results.

Review text

Singularities Common ... section.

1. The term singularities is not defined. Changing singularities to invariant sets seems more appropriate and more precise. In addition, first time readers might confuse this with the critical set where the Jacobian matrix is singular. The terms in this section are (or should be) already defined in other Scholarpedia articles: fixed points, periodic points, stable/unstable manifolds. and invariant circles. Attractors are mentioned, but attracting sets are not. Attractors and attracting sets should be defined before basin of attraction and then basin boundary, although just listing these invariant sets (with links) and saving more detail for the phenomena specific to noninvertible sets seems appropriate. Existing Scholarpedia articles by Milnor (attractors) and Ott (basin of attraction) already exist and provide good definitions of these terms. (Note: The term repelling is usually used as a synonym for expanding. Saddles are unstable, but not repelling.)

Answer

The term "singularities" is the common one used in the famous Andronov' school of nonlinear dynamics in the framework of the qualitative methods (see the book of E. S. Boïko "The Academician A. A. Andronov' school". (in Russian). Ed. Nauka, Moscow, 1983). From 1958, beginning of my researches, the publications of this unequalled school gave me access to an exceptionally wide information, rather unknown in western countries at that time. This unawareness occurred in spite of a text due to J.P. La Salle and S. Lefschetz (J. of Math. Anal. and Appl., 2, pp. 467-499, 1961), who wrote in 1961 :

"In USSR the study of differential equations has profound roots, and in this subject the USSR occupies incontestably the first place. One may also say that Soviet specialists, far from working in vacuum, are in intimate contact with applied mathematicians and front rank engineers. This has brought great benefits to the USSR and it is safe to say that USSR has no desire to relinquish these advantages."

Moreover the last part of the article section 1 defines the corresponding approach, which is that of the qualitative methods of nonlinear dynamics, secs. 2 and 3 showing what is understood by "singularity" in the invertible case and in the noninvertible one. Changing singularities to invariant sets does not work, because a critical set is also a singularity (for the solution) interacting with invariant sets, and leading to new bifurcation types. Ditto for the singularities focal points and prefocal sets for map (and inverse maps) having a vanishing denominator. Only the vocabulary adopted is compatible with these different situations.

About the critical set (third sentence), it is not the set where the Jacobian determinant is zero for smooth maps. It is the rank-one image of this set, separating regions with different numbers of rank-one preimages (see in sec. 1, the end of the paragraph dealing with the Myrberg's map). With respect to the basic publications of the French school of iteration (for example see Julia and Fatou: the critical point is the rank-one image of the point where the map derivative is equal to zero), Jacobian equal zero as critical set is an abuse. Moreover with the hypotheses given for the map, it is reminded that the map can also be simply continuous. So the 1980, 1987, 1994 books give many examples with continuous piecewise linear maps. The article section 3 discriminates the smooth cases and the piecewise smooth cases.

For "attracting set" see the addition in the section 5, and the attracting set changed into attractor in section 3.

For repelling see the addition in sec.2.

Review text

• Section 3.1 Definitions and some properties. 1. It is not clear what `Exceptionally non critical lines in the above sense' means

Answer

See in sec. 3 an explanation of this case in the one-dimensional case, and the link with example 5 in subpage of maps with vanishing denominators for the two-dimensional case.

Review text

2. A cycle with +1 and -1 multipliers is not unique to noninvertible maps. It occurs in two-dimensional orientation-reversing maps. Examples of codimension-two bifurcations unique to noninvertible maps are given in [S] and [KOP].

Answer

This part has been cancelled.

Review text

• Figure 3 and accompanying text in Section 3.2: Basins and their bifurcations: 1. It is not clear which `strict inclusion' is being referred to.

Answer

For Fig. 3 it is \(H_0\) which is contained in the region \(Z_2\) as mentioned in sec. 3.

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2. The dynamics of the map would be clearer if the Attractor A were either included and labeled, or at least described.

Answer

Voluntarily A is not defined for Fig. 3. As indicated in the example 2 of the subpage, it can be a fixed point, a cycle, an invariant closed curve, a chaotic attractor.

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3. Change `turn into a points set' to `turn into a set of points'.

Answer

Corrected.

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• Section 4: Absorbing and chaotic areas

1. Provide precise definitions of absorbing area and chaotic area.

Answer

At the level of an entry of encyclopedia a rough definition can be only given, because otherwise this would imply long developments given in the Mira et al. [1996] (a whole chapter)

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• First subpage: 1. Example is misspelled

Answer

Corrected

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2. It would be instructive to know which bifurcations are unique to noninvertible maps.

Answer

All the bifurcations related to example 2, being related to the contact of the critical curve with the basin boundary, are clearly specific to noninvertible maps.