  The chaotic sea below the lowest energy spanning curve of the complete
Fermi-Ulam model (FUM) is numerically investigated when the amplitude of
oscillation 'epsilon' of the moving wall is small. We use scaling analysis near
the integrable to non-integrable transition to describe the average energy as
function of time t and as function of iteration (or collision) number n. If t
is employed as independent variable, the exponents related to the energy
scaling properties of the FUM are different from the ones of a well known
simplification of this model (SFUM). However, if n is employed as independent
variable, the exponents are the same for both FUM and SFUM. In the collision
number analysis, we present analytical arguments supporting that the exponents
c* and b* related to the initial velocity and to 'epsilon' are given by c* =
-1/2 and b* = -1. We derive also a relation connecting the scaling exponents
related to the variables time and collision number. Moreover, we show that,
differently from the SFUM, the average energy in the FUM saturates for long
times and we justify the physical origins for some differences and similarities
observed between the FUM and its simplification.
