Deceleration parameter as a unifying criterion for FRW cosmological models at critical points

The critical points of dynamical system equations are key in understanding the time evolution of the relevant model-universe. They provide information which help us to rule out those cosmological models which are not able to correctly describe the evolution of the universe from unstable epochs of its early-time to stable points in the distance future. Having the conventional $\frac{\dot{H}}{H^2}=ϵ({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_k)$ where $({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_k)$ are dynamical-system variables, here, we show that it is only the deceleration parameter that is important for FRW cosmological models at the level of the critical points. Meaning that, if the deceleration parameter $q_c$ is known at any critical point of the universe, then other cosmographic parameters (which are related to high derivatives of scale factor $a$) will be determined automatically in terms of the deceleration parameter as having $\frac{1}{H^n}\frac{1}{a}\frac{d^n\left(a\right)}{d{t}^n}={\prod }_{j=1}^n((2-j)+(1-j)q_c)$. This indicates that any critical point of FRW models can be fully described in terms of the deceleration parameter alone when other cosmographic parameters (in relation to higher derivatives of $a$) are obtained as polynomial functions of $q_c$, which hints at a sort of universality in the cosmic evolution for these epochs.