  We take a central spin model (CSM), consisting of a one dimensional
environmental Ising spin chain and a single qubit connected globally to all the
spins of the environment, to study numerically the excess energy (EE) of the
environment and the logarithm of decoherence factor namely, dynamical fidelity
susceptibility per site (DFSS), associated with the qubit under a periodic
driving of the transverse field term of environment across its critical point
using the Floquet technique. The coupling to the qubit, prepared in a pure
state, with the transverse field of the spin chain yields two sets of EE
corresponding to the two species of Floquet operators. In the limit of weak
coupling, we derive an approximated expression of DFSS after an infinite number
of driving period which can successfully estimate the low and intermediate
frequency behavior of numerically obtained DFSS. Our main focus is to
analytically investigate the effect of system-environment coupling strength on
the EEs and DFSS and relate the behavior of DFSS to EEs as a function of
frequency by plausible analytical arguments. We explicitly show that the
low-frequency beating like pattern of DFSS is an outcome of two frequencies,
causing the oscillations in the two branches of EEs, that are dependent on the
coupling strength. In the intermediate frequency regime, dip structure observed
in DFSS can be justified by the resonance peaks of EEs at those coupling
parameter dependent frequencies; high frequency saturation value of EEs and
DFSS are also connected to each other by the coupling strength.
