  In this paper, we have introduced a numerical method for solving the
Friedmann equations of modified $f(\mathcal{G})$ gravity in the presence of
pressureless matter, enabling us to predict the redshift behavior of the Hubble
expansion rate. To evaluate the credibility of the model, we applied a Bayesian
MCMC technique using late-time cosmic observations to impose limitations on the
free parameters of the theory. Our results suggest that the $f(\mathcal{G})$
model can reproduce the low-redshift behavior of the standard $\Lambda$CDM
model, but there are significant differences at high redshifts, leading to the
absence of a standard matter-dominated epoch. We also examined the profiles of
cosmographic parameters using the model parameter values from the standard
range to verify the intermediate epochs. Our analysis shows that the highly
promising $f(\mathcal{G})$ model is a feasible candidate for explaining the
current epochs. We presented a dynamical system analysis framework to examine
the stability of the model. Our study identified critical points depicting
various phases of the Universe and explained the evolutionary epochs. We
demonstrated that the model effectively captures the evolution of energy
components over cosmic time, supporting its validity as an alternate
explanation for the observed acceleration of the Universe.
