We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height $h$. For chaotic orbits below the periodic islands, the survival probability for the particles to reach $h$ is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when $h$ reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when $h$ reaches the position of the lowest KAM island.

• Received 23 October 2012

DOI:https://doi.org/10.1103/PhysRevE.87.062904