  The stable critical points and their corresponding cosmology are derived in
the teleparallel gravity with an added Gauss-Bonnet topological invariant term.
We have analyzed the dynamics of the Universe by presenting two cosmological
viable models, showing the potential to describe different phases of the
evolution of the Universe. The value of the deceleration parameter ($q$), total
equation of state parameter ($\omega_{tot}$) and dark energy equation of state
parameter ($\omega_{DE}$) have been presented against each critical point. The
existence and stability conditions are also presented. We study the behavior of
the phase space trajectories at each critical point. Finally, the evolutionary
behavior of the deceleration parameter and the equation of state parameters
have been assessed with the initial condition of the dynamical variables, and
compatibility has been observed in connection with the present cosmological
scenario.
