  Fractional cosmology modifies the standard derivative to Caputo's fractional
derivative of order $\mu$, generating changes in General Relativity. Friedmann
equations are modified, and the evolution of the species densities depends on
$\mu$ and the age of the Universe $t_U$. We estimate stringent constraints on
$\mu$ using cosmic chronometers, Type Ia supernovae, and joint analysis. We
obtain $\mu=2.839^{+0.117}_{-0.193}$ within the $1\sigma$ confidence level
providing a non-standard cosmic acceleration at late times; consequently, the
Universe would be older than the standard estimations. Additionally, we present
a stability analysis for different $\mu$ values. This analysis identifies a
late-time attractor corresponding to a power-law decelerated solution for $\mu
< 2$. Moreover, a non-relativistic critical point exists for $\mu > 1$ and a
sink for $\mu > 2$. This solution is a decelerated power-law if $1 < \mu < 2$
and an accelerated power-law solution if $\mu > 2$, consistent with the mean
values obtained from the observational analysis. Therefore, for both flat FLRW
and Bianchi I metrics, the modified Friedmann equations provide a late cosmic
acceleration under this paradigm without introducing a dark energy component.
This approach could be a new path to tackling unsolved cosmological problems.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics