Dynamical properties of a dissipative discontinuous map: A scaling investigation

•We study the effects of dissipation on the scaling properties of nonlinear discontinuous maps.•We investigate the behavior of the average squared action as a function of the nonlinearity and dissipation.•We concentrate our efforts to study the case where the nonlinearity is large.•We prove that dissipation produces an exponential decay for the average action.

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action 〈I2〉〈I2〉 as a function of the n-th iteration of the map as well as the parameters K and γ  , controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., K≫1K≫1. In this regime and for large initial action I0≫KI0≫K, we prove that dissipation produces an exponential decay for the average action 〈I〉〈I〉. Also, for I0≅0I0≅0, we describe the behavior of 〈I2〉〈I2〉 using a scaling function and analytically obtain critical exponents which are used to overlap different curves of 〈I2〉〈I2〉 onto a universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action ω.