  We study the 1D Hamilton systems and their statistical behaviour, assuming
the initial microcanonical distribution and describing its change under a
parametric kick, which by definition means a discontinuous jump of a control
parameter of the system. Following a previous work by Papamikos and Robnik (J.
Phys. A: Math. Theor. 44 (2012) 315102) we specifically analyze the change of
the adiabatic invariant (the action) of the system under a parametric kick: A
conjecture has been put forward that the change of the action at the mean
energy always increases, which means, for the given statistical ensemble, that
the entropy in the mean increases. By means of a detailed analysis of a great
number of case studies we show that the conjecture largely is satisfied, except
if either the potential is not smooth enough, or if the energy is too close to
a stationary point of the potential (separatrix in the phase space). Very fast
changes in a time dependent system quite generally can be well described by
such a picture and by the approximation of a parametric kick, if the change of
the parameter is sufficiently fast and takes place on the time scale of less
than one oscillation period.
