  We develop a general geometric method to establish the existence of positive
Lyapunov exponents for a class of skew products. The technique is applied to
show non-uniform hyperbolicity of some conservative partially hyperbolic
diffeomorphisms having as center dynamics coupled products of standard maps,
notably for skew-products whose fiber dynamics is given by (a continuum of
parameters in) the Froeschl\'e family. These types of coupled systems appear as
some induced maps in models for the study of Arnold diffusion.
  Consequently, we are able to present new examples of partially hyperbolic
diffeomorphisms having rich high dimensional center dynamics. The methods are
also suitable for studying cocycles over shift spaces, and do not demand any
low dimensionality condition on the fiber.

Dynamical Systems