Dynamical properties of a dissipative discontinuous map: A scaling investigation

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action $\left< I^2 \right>$ as a function of the $n$-th iteration of the map as well as the parameters $K$ and $\gamma$, controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., $K\gg 1$. In this regime and for large initial action $I_0\gg K$, we prove that dissipation produces an exponential decay for the average action $\left< I \right>$. Also, for $I_0\cong 0$, we describe the behavior of $\left< I^2 \right>$ using a scaling function and analytically obtain critical exponents which are used to overlap different curves of $\left< I^2 \right>$ onto an universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action $\omega$.