  In this paper, we derive the field equations of modified Gauss-Bonnet gravity
termed as $f(R,G)$ gravity for the non-flat Friedmann-Robertson-Walker (FRW)
spacetime. We utilize the dynamical system approach to study the cosmic
dynamics of two different class of $f(R,G)$ models composed of radiation and
matter (cold dark matter and baryonic matter). The linear perturbations around
the fixed points are studied to explore the corresponding stability of points.
The cosmological implications are studied in $f(R,G)=f_0R^nG^{1-n}$ and
$f(R,G)=f_0R^\alpha+f_1G^\beta$ models to identify the qualitative evolution of
universe with the flat-FRW spacetime. The qualitative differences between the
considered class of models are discussed in detail. The fixed points
corresponding to the late-time accelerated and radiation phase of the universe
will exist in the model but, the existence of fixed point corresponding to the
matter dominated phase will depend on the functional form of $f(R,G)$.
Furthermore, the autonomous systems are utilized to study the cosmographic
parameters along with the statefinder diagnostic.
