View source for Talk:Noninvertibility

This article is good as is, but a slightly expanded version might make the article 
more useful to a wider audience, especially to nonexperts.
Some wording and ordering changes are suggested.  These are largely to 
more completely set the context of noninvertibility and to make the development as
parallel as possible.

General comments/suggestions:
<ol>
<li>The article should mention the best known examples of 
noninvertibility: the logistic/quadratic map in 1D, and the complex
quadratic family z -> z<sup>2</sup> + C.
Even though the details about these maps are best left to articles on those specific
topics, it would better place the topics treated in this article in context.
<li> Along the same lines, although the current version of the article refers to
C. Mira's Scholarpedia article on Noninvertible Maps,
more emphasis could be given to the pioneering work of Mira and 
his colleagues on planar endomorphisms.  Details should still be left to
Mira's article, but this would better establish the relationship between
Mira's article and the current one.
<li> There seems to be a conspicuous lack of references to PDEs exhibiting nonlinear phenomena.
It would be nice to know if there are such references.  If not,
then it should be clear that the PDEs are mentioned only because they fall into the class
of noninvertible systems.
</ol>

<br>
Specific comments/suggestions:
<ol>
<li>Definition.
It seems more natural to start with a definition of an invertible dynamical
system.  A noninvertible dynamical system is then one which is not invertible.

The definition of (non)invertible seems to be more straightforward for maps.
Interval of existence seems to cause possible problems with a definition
for ODEs. For example, x'=x<sup>2</sup> is not defined for all forward times.
Similarly, the logistic ODE: x'=x(1-x) is not defined for all backward times,
but most mathematicians consider it an invertible dynamical system,
at least in its interval of existence.
The existence-uniqueness theorem of ODEs guarantees
that all ODEs which satisfy its hypotheses are locally invertible.
Some discrete maps have the same existence issue,
for example when the forward or backward map is defined with a denominator which
can be zero. Maybe interval of existence needs to be part of the definition.
(This obviously does not resolve this issue.)

<li> Paragraph 1.

Saying that "different initial states lead to different dynamics" might be
misleading, since we usually say, even for invertible systems, that
solutions with the same long-term fate display the "same dynamics."
It might be better to say "two different initial states display different
states, both forward and backward in time."

<li> Paragraph 2.
Reorder to present mathematical theory, then applications, then natural
processes.  (Natural processes are mentioned, but no specific example is
given, so I'm not sure whether this should be expanded or deleted.)


<br>
NOTE: The link to "iterated maps" in "invertible {iterated maps}"  is to a
demonstration for the (noninvertible) logistic map.

<li>Noninvertibility in bifurcations and chaos section.  Make organization more
parallel: generally bifurcations, chaos, then bifurcation routes to chaos.

<ol>
<li> Finite dimensional maps
 <ol>
<li>1D maps: bifurcations: homoclinic, choatic attractors, period-doubling route to chaos,
role of critical orbit (references)
 <li> 1D complex maps: bifurcations: beteween bulbs, inside to outside the Mandelbrot set, 
chaos on Julia sets, role of critical orbit (references)
 <li> 2D real maps: J_0, J_k, bifurcations involving the interaction of dynamical objects
with J_k (fixed/periodic orbits, unstable manifolds (loop formation),
invariant circles (loop formation and breakup of invariant circles, especially in
a "noninvertible route to chaos"), Gumowski-Mira pioneering work, absorbing regions,
holes in basins, contact bifurcations (attractor with its basin boundary),
codimension-two cusp-cusp unfolding
</ol>

<li>Infinite dimensional systems 
 <ol> 
<li>Delay Differential Equations
<li> PDEs
</ol>

Here are some questions I think a Scholarpedia reader might want to know:
<ol>

<li>In the infinite dimensional systems, are there standard codimension-one bifurcations,
chaos, route(s) to chaos?
<li>Is there an analogue of J_0?
<li> Are all the features of noninvertible maps possible in infinite-dimensional systems?
Are there features that are not possible in two-dimensional maps?  What about vice versa? 
What about bifurcations?
</ol>
</ol>
</ol>

Possible References to add:

<ul>
<li>Abraham R., Gardini L., Mira C. [1997] "Chaos in Discrete Dynamical Systems. A visual introduction in two dimensions" (Telos, the Electronic Library of Science, Springer-Verlag, New York) (book/CD-ROM package).
<li>Gumowski I. & Mira C. [1980] "Recurrences and discrete dynamic systems - An introduction".
250 pages. Lecture notes in mathematics n° 809, Springer.
<li>Mira C., Gardini L., Barugola A. & Cathala J.C. [1996] "Chaotic Dynamics in
two-dimensional noninvertible maps". World Scientific, "Series A on Nonlinear Sciences",
vol.20, 622 pages.
<li>Collet and Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhauser, 1980.
[Devaney 1992] Devaney, A First Course in Chaotic Dynamical Systems. Now published by
Westview Press, 1992.
<li>Devaney R., An Introduction to Chaotic Dynamical Systems
Second Edition. Now published by Westview Press, 2003. 
<li>Sander E, "Homoclinic tangles for noninvertible maps,"  Nonlinear Analysis, 41(1-2):259-276, 2000.
<li>Any other relevant Scholarpedia references.
</ul>