  In recent years, the modified theory of gravity known as $f(Q)$ gravity has
drawn interest as a potential alternative to general relativity. According to
this theory, the gravitational force is determined by a function of the
so-called ``non-metricity" tensor $Q$, which expresses how far a particle
space-time is from the metric geometry. In contrast to general relativity,
which describes the gravitational field using the curvature tensor, $f(Q)$
gravity builds a theory of gravity using the non-metricity tensor. For this
class of theories, dynamical system analysis of the background and perturbation
equations has been carried out in this work to determine how various models
behave cosmologically. Here, the critical points are determined for two $f(Q)$
models from the literature: the power law, $ f(Q)=Q+mQ^{n} $, and the
logarithmic, $ f(Q)=\alpha+\beta log Q $ models. The stability behaviour and
corresponding cosmology are displayed for each critical point. For the power
law model, we achieve a matter-dominated saddle point with the right matter
perturbation growth rate. For the logarithmic model, we get a saddle point
dominated by the geometric component of the $ f(Q)$ model with perturbations in
the decomposition of matter. For both models, we later achieved a stable and
accelerating Universe with constant matter perturbations.
