  The higher-curvature gravity with boundary terms i.e the $f(Q)$ theories,
grounded on non-metricity as a fundamental geometric quantity, exhibit
remarkable efficacy in portraying late-time universe phenomena. The aim is to
delineate constraints on two prevalent models within this framework, namely the
Log-square-root model and the Hyperbolic tangent-power model, by employing the
framework of Big Bang Nucleosynthesis (BBN). The approach involves elucidating
deviations induced by higher-curvature gravity with boundary terms in the
freeze-out temperature ($T_{f}$) concerning its departure from the standard
$\Lambda$CDM evolution. Subsequently, constraints on pertinent model parameters
are established by imposing limitations on $\vert \frac{\delta
T_{f}}{T_{f}}\vert$ derived from observational bounds. This investigation
employs dynamical system analysis, scrutinizing both background and perturbed
equations. The study systematically explores the phase space of the models,
identifying equilibrium points, evaluating their stability, and comprehending
the system's trajectory around each critical point. The principal findings of
this analysis reveal the presence of a matter-dominated saddle point
characterized by the appropriate matter perturbation growth rate. Subsequently,
this phase transitions into a stable phase of a dark-energy-dominated,
accelerating universe, marked by consistent matter perturbations. Overall, the
study substantiates observational confrontations, affirming the potential of
higher-curvature gravity with boundary terms as a promising alternative to the
$\Lambda$CDM concordance model. The methodological approach underscores the
significance of dynamical systems as an independent means to validate and
comprehend the cosmological implications of these theories.
