Leaking of orbits from the phase space of the dissipative discontinuous standard mapping

Abstract

We investigate the escape of particles from the phase space produced by a two-dimensional, nonlinear and discontinuous, area-contracting map. The mapping, given in action-angle variables, is parametrized by K and γ which control the strength of nonlinearity and dissipation, respectively. We focus on two dynamical regimes, K<1 and K≥1, known as slow and quasilinear diffusion regimes, respectively, for the area-preserving version of the map (i.e., when γ=0). When a hole of hight h is introduced in the action axis we find both the histogram of escape times PE(n) and the survival probability PS(n) of particles to be scale invariant, with the typical escape time ntyp=exp(lnn); that is, both PE(n/ntyp) and PS(n/ntyp) define universal functions. Moreover, for γ≪1, we show that ntyp is proportional to h2/D, where D is the diffusion coefficient of the corresponding area-preserving map that in turn is proportional to K5/2 and K2 in the slow and the quasilinear diffusion regimes, respectively.