
Using an argument of Aubry, there is an elegant proof of space time chaos in
coupled map lattices. (See the paper with A. Jakobsen in Physics letters A).
These dynamical systems are discrete partial difference equations.
The pictures shows an example of an orbit of length 800 of a Hamiltonian coupled map lattice. 400 Standard maps are coupled in the following way, where n is the spatial coordinate taken modulo 400 and x(n),y(n) are taken modulo 1:
If g is large, the Standard map has horseshoes, compact invariant sets on which the dynamics is conjugated to a Markov chain. This stays true for the coupled maps in finite or infinite dimensions. 
