Wikipedia has a nice linked list to begin with:

http://en.wikipedia.org/wiki/List_of_chaotic_maps

Dear Artur:

Thanks for this list - it is a good start to understand this topic, but does not provide satisfactory information about the discrete chaotic maps. For instance, many of the chaotic maps do not have individual Wiki links and for the maps with Wiki links, I see only properties and details about "Lyapunov exponents" are missing.

I am looking for a book or a good survey paper on discrete chaotic maps with details of Lyapunov exponents & Kaplan-Yorke dimension.

Thank you!

Best wishes,

Sundar

The literature on chaotic maps is quite large. Good general introductions to chaotic maps can be found in texts books like:

"Chaos: An Introduction to Dynamical Systems" by Kathleen T. Alligood, Tim Sauer, and James Yorke

"Chaos in Dynamical Systems" by Edward Ott

More specialized presentations can be found in:

"Iterated Maps on the Interval as Dynamical Systems" by Pierre Collet, J.-P. Eckmann

"Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension" by Abraham Boyarsky

In the most books the maps, particularly, in Hamiltonian systems are presented as  models and they are not rigorously derived. It turns out the known methods used for this purpose are simply incorrect, and they lead, in some case, to wrong results (see e.g.,  S.S. Abdullaev, Phys. Rev. E, v. 76, 026216 (2007)). The problem is that the variables of maps do not coincide with the variables of the original continuous system. Only recently this problem has been resolved in Hamiltonian systems by developing  the rigorous method to derive maps from continuous systems. The method has been thoroughly described in my book "Construction of mappings in Hamiltonian systems and their applications" (Springer, 2006). Its contents can be found in my site of Research Gate. The application of maps in plasma physics is also discussed in my recent book "Magnetic stochasticity in magnetically confined fusion plasmas" (Springer, 2014).   

Professor Abdullaev:

Thank you for the kind reply. My thesis guide, (Late) Dr. Chris Byrnes, was also very interested in Hamiltonian systems. I shall read your works!

Best wishes,

Sundar

One recent application of maps (most of them having chaotic behavior in some parameter region) is describing neurons in computational simulations of neuronal systems. Although no detailed description of the chaotic behavior is available, you can check some discussion in our paper (and references there in):

https://www.researchgate.net/publication/255690486_A_brief_history_of_excitable_map-based_neurons_and_neural_networks

Some other papers of maps having chaotic behavior:

1996: http://www.worldscientific.com/doi/abs/10.1142/S0218127496001508

2002: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.65.041922

2007: http://scitation.aip.org/content/aip/journal/chaos/17/4/10.1063/1.2795435

Article A Brief History of Excitable Map-Based Neurons and Neural Networks

Finding chaos in nonlinear iterated maps is too trivial of a problem to be of much interest. My book Strange Attractors: Creating Pattern in Chaos lists over 350 examples and provides software for producing thousands more. The book (from 1993) is now out of print but is freely available in pdf form on the Web at http://sprott.physics.wisc.edu/fractals/booktext/sabook.pdf.

Professor Sprott:

Thanks a lot for your reply & reference.

Best wishes,

Sundar

the book of prof E. Zeraoulia (who  works very often with Prof J. C.  Sprott)

http://www.crcpress.com/product/isbn/9781466580701

gives one thousand references on Hénon and Lozi map, with very theoretic studies and new applications. Nowadays it is interesting to note that the main interest of chaotic maps lies in Evolutionary Algorithms, Genetic algorithms,  chaotic optimization and chaots based cryptography. Hundred of papers are published in thes fields. Read for example: http://link.springer.com/chapter/10.1007%2F978-3-642-41968-3_38#page-1

or Google prof Ivan Zelinka Ostrava university who is the specialist for E.A.,

and check prof Safawan El Hassad on RG

https://www.researchgate.net/profile/Safwan_El_Assad2

Best wishes

René Lozi

Dear René Pierre Lozi :

Thanks for providing me good references on chaotic maps!

Best wishes

Sundar

Others here have offered excellent commentary and insight.

A few years ago my colleagues and I published an article in the Bulletin of Mathematical Biology that detailed various phase-space outcomes in a model of two interacting blood-stem cells. Our work utilized discrete parameter changes over time, although we did not work with a strange or chaotic attractor.

The article is"Dynamic stabilization in the PU1-GATA1 circuit using a model with time-dependent kinetic change" and is on my Research Gate page.

Dear Dr. Vaidyanathan,

For the noninvertible maps class, the Scholarpedia article

- "Noninvertible maps" http://www.scholarpedia.org/article/Noninvertible_maps gives a basic information

• Also see 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).

• The book: 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) is a treatise on the topic, illustrated by many examples, about the fundamental properties introduced by the map noninvertibility.

A developed information on invertible, or noninvertible maps, is given by the 3 books

• Gumowski I. & Mira C. [1980a] "Dynamique chaotique. Transformations ponctuelles. Transition ordre-désordre. 480 pages. Cépadues Editions, (1980), Toulouse.

• Gumowski I. & Mira C. [1980b] "Recurrences and discrete dynamic systems - An introduction". 250 pages. Lecture notes in mathematics n° 809, Springer.

• Mira C. [1987] "Chaotic Dynamics. From the one-dimensional endomorphism to the two-dimensional diffeomorphism". World Scientific, 449 pages.

- For the class of invertible, or noninvertible maps with vanishing denominator, the following Scholarpedia article gives a basic information http://www.scholarpedia.org/article/Maps_with_vanishing_denominators

A survey of results obtained on nonlinear maps from the year 1963, by the Toulouse research group, is given via the paper "Nonlinear maps: from the Toulouse colloquium (1973) to NOMAD'13". The full text is obtained via the attached file and  the Research Gate reference:

https://www.researchgate.net/publication/263889910_Nonlinear_maps__From_the_Toulouse_colloquium_%281973%29_to_NOMA%2713?ev=prf_pub

An extract of the above [1980b] book is given in the Research Gate reference "About what is called Gumowski-Mira map. Mathematical properties"

https://www.researchgate.net/publication/263927808_ABOUT_WHAT_IS_CALLED_GUMOWSKI-MIRA_MAP._%282%29_MATHEMATICAL_PROPERTIES?ev=prf_pub

Sincerely

Christian Mira

Conference Paper Nonlinear maps: From the Toulouse Colloqium (1973) to NOMA’13

Technical Report ABOUT WHAT IS CALLED "GUMOWSKI-MIRA MAP". (2) MATHEMATICAL PROPERTIES

Probably will be useful:

Y.-C. Lai, T. Tél Transient Chaos, Complex Dynamics on Finite-Time Scales, Springer, 2011.

Li Z. Fuzzy Chaotic Systems. Modeling, Control, and Applications. Springer, 2006.