There is a class of dynamical systems, called trace maps, which are polynomial systems acting on (real and complex) analytic manifolds. If you google "Fibonacci Trace Map", you'll find a few good links. The amazing thing about these systems is that they exhibit quite a few nontrivial dynamical phenomena: Axiom A, partial hyperbolicity, and Newhouse phenomena (the latter paper is still in preparation). In particular, Newhouse phenomena leads to coexistance of a chaotic sea and elliptic islands, and the chaotic sea is of full Hausdorff dimension (ideally, one would want to show positive measure of the chaotic sea, of course).

Amazingly, these systems first appeared very naturally in physics in early 1980's (see [1] and [2] below). They arise in quite a few fields: algebra, Painleve equations (see [3] below), Ising models and kicked two-level systems (see [4] below), and spectral theory of quantum energy Hamiltonians where the medium is a class of matter called quasi-crystals (yes, that same thing that Dan Shechtman got the 2012 Nobel prize in chemistry for) (see [1] and [2] for the classics and [5] and [6] for recent work).

I should also mention quite a few works in the 3-body problem where hyperbolic dynamics appear, but I am not an expert in the field, so can't really point out any good references. The only paper that comes to mind (because I had looked at it previously) is http://www2.math.umd.edu/~vkaloshi/papers/HD-Sept2011.pdf.

By the way, I would be very happy to find "real" applications of nonuniformly hyperbolic dynamics, but I have a good feeling that those systems would probably arise in nonequillibrium statistical mechanics (in which I'm not very knowledgeable).

[1] M. Kohmoto, L. P. Kadanoff, and C. Tang. Localization problem in one dimension: Mapping and escape. Phys. Rev. Lett., 50(23):1870–1872, 1983.

[2] S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. D. Siggia. One-dimensional Schrodinger equation with an almost periodic potential. Phys. Rev. Lett., 50(23):1873–1876, 1983.

[3] S. Cantat. Bers and Henon, Painleve and Schrodinger. Duke Math. J., 149(3):411–460, sep 2009.

[4] M. Baake, U. Grimm, and D. Joseph. Trace maps, invariants, and some of their applications. Int. J. Mod. Phys. B, 7(06–07):1527–1550, 1993.

[5] http://arxiv.org/find/math/1/au:+Gorodetski_A/0/1/0/all/0/1 (anything that involves the word "Fibonacci" or "Quasi..." in the title)

[6] http://arxiv.org/find/math/1/au:+Yessen_W/0/1/0/all/0/1 for applications to other models.