Q:

Self Study in Dynamical Systems

I'm trying to get into the field of dynamical systems by (self) studying one-dimensional dynamics and circle homeomorphisms; for my guidance, I'm trying to assemble materials in this field that obey the following:

Historical aspects (optional);
A good topological approach;
Without use Mathematica; for the numerical part, I'd like to use a more friendly programming language (that sounds quite subjective) like C, Python, FORTRAN, Perl, etc.
Make a natural transition to more advanced topics (my goal is to get the road to topological dynamics); 

I haven't included this one in my list (because I'm very sure that requires advanced treatment), but I'd love to get access to materials that relates dynamical systems to number theory, set theory and mathematical logic. 
I have a background in abstract and linear algebra (in the level of Jacobson's Lectures in Algebra and Halmos's Finite Dimensional Vector Spaces), in general topology (Munkres's Topology, make transition to Engelking's General Topology) and set theory (Jech's Introduction to Set Theory).

A:

Good books with enough introductory material in this regard are:

introduction to dynamical systems, Michael Brin.

This book defines all the basic concepts of dynamic requirements of Chapter one is called
Low dimensional dynamics
Where does the topological classification of homeomorphisms of the circle (Theorem of Poincare). It also has the interesting theorem Sharkovsky (Generalization of period three implies chaos).
2.A First Course in Dynamics with a Panorama of Recent Developments.
This book covers a similar content, but with a different style of writing.
For books that go deeper into the core of the theory are two books by the same author:

Lectures on One-Dimensional Dynamics. Publ. do 17º Colóquio Brasileiro de Matemática, 1988
One Dimensional Dynamics. Springer-Verlag, 1993 
With S. Van Strien

You can visit the author page:   http://w3.impa.br/~demelo/listapub.html
Regarding prerequisites believe that in either case you need to know something of Differential Calculus.
When you want to discuss one-dimensional dynamic write to me.

A:

Holmgren's "A First Course in Discrete Dynamical Systems" is very good written with simple worlds book. At the end of the book author gives list of books which are good continuation of one he wrote. 

A:

Check out math video lectures among other subjects on Chaotic Dynamical systems  UCCS Math Video Lectures 

