Diffusion by extrinsic noise in the kicked Harper map

Abstract

A significantly improved analytic understanding of the extrinsically driven diffusion process is presented in a nonlinear dynamical system in which the phase space is divided into periodic two-dimensional tiles of regular motion, separated by a connected separatrix network (web) [previously studied by A. J. Lichtenberg and Blake P. Wood, Phys. Rev. Lett. 62, 2213 (1989)]. The system is represented by the usual "kicked Harper map" with added extrinsic noise terms. Three different diffusion regimes are found depending upon the strength of the extrinsic perturbation I relative to the web and regular motions. When the extrinsic noise is dominant over the intrinsic stochasticity and the regular rotation motions in the tile, diffusion obeys the random phase scaling t(2) When the extrinsic noise is dominant over the intrinsic stochasticity, but weaker than the regular rotation motion, the diffusion scales as lK(1/2) where K is the strength of the intrinsic kick. These findings agree well with numerical simulation results. When the extrinsic noise process is weaker than the stochastic web process, we analytically reproduce the well-known numerical result: The web diffusion is reduced by the ratio of phase-space areas of intrinsic to extrinsic stochasticity.