14
$\begingroup$

I have just begun my first dynamical systems class, and I would like to try out the advice in the top answer here. To summarize, the answer suggests that when studying a new field, one should look at the original papers, and supplement it with modern material.

My class is roughly following Introduction to Dynamical Systems by Brin and Stuck. We are starting in the second chapter, which is about topological dynamics (it is only 20 pages, but very dense). I was wondering if you could please recommend me some authors/papers which started the field?

Thank you very much.

$\endgroup$
2
  • 3
    $\begingroup$ Smale’s 1967 “differentiable dynamical systems” is one highly influential example. You might also want to have a look at the book by Guckenheimer and Holmes. Conley’s “isolated invariant sets and the Morse index” is one example of an influential work in topological dynamics specifically. $\endgroup$ – Matthew Kvalheim Sep 4 '20 at 23:38
  • 1
    $\begingroup$ The geometrical approach was initiated by Poincare , in ' les méthodes nouvelles de la mécanique céleste'. $\endgroup$ – Piyush Grover Sep 5 '20 at 3:36
11
$\begingroup$

Philip Holmes has summarized on Scholarpedia the seminal early developments of the field of dynamical systems, from the mathematical point of view (which I understand is the view point of the OP). Starting from the classic works of Poincaré and Birkhoff, through Andronov, Pontryagin and the Moscow School, to the "era of chaos" (Smale, Levinson, Chirikov, Lorenz, KAM, Ruelle & Takens, ...).

$\endgroup$
5
$\begingroup$

I would suggest to concentrate on papers that introduce interesting new dynamical systems.

Smale's 1967 paper "differentiable dynamical systems" has been mentioned in the comments. Actually everything written by Smale is highly recommended. Smale's horseshoe appears in his 1963 pretty short article "A structurally stable differentiable homeomorphism with an infinite number of periodic points." See Smale, Steve "Finding a horseshoe on the beaches of Rio". Math. Intelligencer 20 (1998), no. 1, 39–44 for a short account of his discovery.

The 1963 paper by Edward Lorenz entitled "deterministic nonperiodic flow" is also quite readable. There he introduced what is now called the Lorenz attractor.

Cvitanovic's 1984 book "universality in chaos" is a collection of interesting articles in dynamical systems published prior to the eighties that is worth browsing. It contains reprints of seminal papers from Ruelle, Feigenbaum, Henon for example. There is another sourcebook compiled by a chinese mathematician that is also nice, unfortunately I can't find the reference right now.

$\endgroup$
2
$\begingroup$

Katok / Hasselblatt: Introduction to the Modern Theory of Dynamical Systems gives a good introduction.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.