There is a well-known "chaotic hypothesis" dating from 1995 or so in statistical physics that suggests that classical statistical-physical systems should be "effectively" Anosov. I won't get into the details of this, but I want to use it (and the observation that despite the Heisenberg group nilmanifold example, from the POV of statistical physics most "useful" Anosov systems are based on toral automorphisms or perturbations thereof) to motivate the following question:

Given a generic "nice" Hamiltonian $H$, at any point it should admit a quadratic approximation and the corresponding toral automorphism (which exists and is unique). Does anyone think that it is plausible to, or better still know of attempts (especially successful ones) to, work on the chaotic hypothesis by piecing these local approximations together (I'm naively thinking of taking connected sums of tori and patching together integral curves, which seems a little silly, but still...)?