  Within the supersymmetry scheme, which includes many-body interactions and a
perturbation possessing the SO(5) (or SU(5)) symmetry, the rotational bands of
the $A\sim 250$ mass region are studied systematically. A novel modification is
introduced, extending the Arima coefficient to the third order. This study is
dedicated to the quantitative analysis of evolving trends in intraband
$\gamma$-transition energy, kinematic, and the dynamic moment of inertia within
the rotational bands of $^{244}$Pu and $^{248}$Cm. The computed outcomes
exhibit an exceptional degree of agreement with experimental observations
across various conditions. The significance of including a higher-order Arima
coefficient is further examined by contrasting it with the previously suggested
model. The calculated results demonstrate the significance of both the
anti-pairing and pairing effects in the evolution of the dynamic moment of
inertia. Additionally, these insights reveal the importance of a newly
introduced parameter in accurately depicting complex nuclear behaviors such as
back-bending, up-bending, and the downturn in the moment of inertia.
