Diffusion in systems with many degrees of freedom will be studied for weakly coupled dynamical mappings and for coupled oscillator chains. Local diffusion across and along resonance layers in action space is used to determine the global diffusion properties of the coupled systems. The diffusion along resonance layers, known as Arnold diffusion, is determined using a three-resonance model. The upper bounds to diffusion and the approach to energy equipartition will be determined as a function of the number of degrees of freedom and the strength of the coupling perturbation. Comparisons of analytic approximations to the numerical results will allow an analytic formalism to be developed that can be extended beyond the range of numerical analysis.
