  To describe fusion hindrance observed in fusion reactions at extremely low
incident energies, I propose a novel extension of the standard CC model by
introducing a damping factor that describes a smooth transition from sudden to
adiabatic processes. I demonstrate the performance of this model by
systematically investigating various deep sub-barrier fusion reactions. I
extend the standard CC model by introducing a damping factor into the coupling
matrix elements in the standard CC model. I adopt the Yukawa-plus-exponential
(YPE) model as a basic heavy ion-ion potential, which is advantageous for a
unified description of the one- and two-body potentials. For the purpose of
these systematic investigations, I approximate the one-body potential with a
third-order polynomial function based on the YPE model. Calculated fusion cross
sections for the medium-heavy mass systems of $^{64}$Ni + $^{64}$Ni, $^{58}$Ni
+ $^{58}$Ni, and $^{58}$Ni + $^{54}$Fe, the medium-light mass systems of
$^{40}$Ca + $^{40}$Ca, $^{48}$Ca + $^{48}$Ca, and $^{24}$Mg + $^{30}$Si, and
the mass-asymmetric systems of $^{48}$Ca + $^{96}$Zr and $^{16}$O + $^{208}$Pb
are consistent with the experimental data. The astrophysical S factor and
logarithmic derivative representations of these are also in good agreement with
the experimental data. Since the results calculated with the damping factor are
in excellent agreement with the experimental data in all systems, I conclude
that the smooth transition from the sudden to adiabatic processes occurs and
that a coordinate-dependent coupling strength is responsible for the fusion
hindrance. In all systems, the potential energies at the touching point $V_{\rm
Touch}$ strongly correlate with the incident threshold energies for which the
fusion hindrance starts to emerge, except for the medium-light mass systems.
