  The manuscript presents the dynamics of Tsallis holographic dark energy
(THDE) and R\'enyi holographic dark energy (RHDE) prescribed by a non-linear
interaction in the FRW spacetime and for a scale factor evolving with a
composite power law-exponential (hybrid) form. To construct the energy
densities of these holographic dark energy models, I assume the Hubble cutoff
to be the IR limit. I find that the deceleration parameter undergoes a
signature flipping at a redshift $z$ consistent with observations. The EoS
parameter $\omega_{de}$ for both the HDE models exhibit quite contrasting
dynamical behavior despite assuming values close to $-1$ at $z=0$ and therefore
consistent with current observations. Next, I find the squared sound speed
$c_{s}^{2}$ to be positive for the THDE model ensuring stability against
perturbations, whereas for the RHDE model, $c_{s}^{2}<0$ implying instability
against perturbations. Furthermore, I analyzed the evolutionary behavior of the
EoS parameter of the HDE models by constructing the
$\omega_{de}-\omega_{de}^{'}$ plane and find that the plane lies in the
freezing region for the THDE model and in the thawing region for the RHDE
model.
