A numerical study of the stabilization effect of steepness

We present the results of numerical experiments about the influence of steepness on the resonant structure, stability and diffusion in a 4-dimensional symplectic map. The map is designed so that by changing a parameter, we smoothly switch steepness on and off by the change of the so called steepness coefficients. In both cases we measure the diffusion coefficients of the actions within a resonance. According to Nekhoroshev theorem we find that, in the steep case, the diffusion coefficients are definitely smaller than in the non steep one, thus confirming the threshold effect of the steepness coefficients which comes from the proof of Nekhoroshev theorem.