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 P_{E}(n) and the survival probability P_{S}(n) of particles to be scale invariant, with the typical escape time n_{typ}=exp〈lnn〉; that is, both P_{E}(n/n_{typ}) and P_{S}(n/n_{typ}) define universal functions. Moreover, for γ≪1, we show that n_{typ} is proportional to h^{2}/D, where D is the diffusion coefficient of the corresponding area-preserving map that in turn is proportional to K^{5/2} and K^{2} in the slow and the quasilinear diffusion regimes, respectively.