Dynamical analysis of a hyperbolic solution in Scale-covariant theory

We study an isotropic flat FLRW-model in Scale-covariant theory of gravity $f_{\gamma \delta}(\phi)$ \cite{Canuto:1977zz} which is explained in terms of ordinary and covariant differentiation of scalar field $\phi$. As we know the deceleration parameter is time-dependent, so we consider the deceleration parameter $q$ as the function of $t$. Using this methodology, we find all the important cosmological factors in terms of a hyperbolic function of the cosmic time $t$. In turn, we create the model having the behavior of the late-time universe, which is ever accelerated expanding and faces a Big Freeze at the end. The model shows the quintessence dark energy model from early to late times. We compute the constrained values of Hubble parameter $H_0=70.979^{+0.021}_{-0.0043}$ and the model parameter $n=1.24079^{+0.00015}_{-0.00079}$ using joint analysis of the $OHD$ data of 77-points and Pantheon bin data. The model exhibits point-type singularity, beginning with a point of zero volume, infinite energy density, and temperature. Furthermore, we obtain the present deceleration parameter $(q_0) \approx {-0.54}$. Also, we examine the ultimate behavior of our model by properly analyzing energy conditions, cosmographical parameters, and Statefinder diagnostic. Finally, the proposed model behaves like a quintessence dark energy model.