What are some foundational authors/papers in dynamical systems?

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, ...).

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.

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