Scaling of dynamical properties of the Fermi–Ulam accelerator

We investigate numerically the chaotic sea of the complete Fermi–Ulam model (FUM) and of its simplified version (SFUM). We perform a scaling analysis near the integrable to non-integrable transition to describe the average energy as function of time tt and as function of iteration (or collision) number nn. When tt is employed as independent variable, the exponents of FUM and SFUM are different. However, when nn is used, the exponents are the same for both FUM and SFUM. In the collision number analysis, we present analytical arguments supporting the values of the exponents related to the control paramenter and to the initial velocity. We describe also how the scaling exponents obtained by using tt as independent variable are related to the ones obtained with nn. In contrast to SFUM, the average energy in FUM saturates for long times. We discuss the origin of the observed differences and similarities between FUM and its simplified version.