  The evolutionary behavior of the Universe has been analysed through the
dynamical system analysis in $f(T,B,T_G,B_G)$ gravity, where $T$, $B$, $T_G$,
and $B_G$ respectively represent torsion, boundary term, teleparallel
Gauss-Bonnet term and Gauss-Bonnet boundary term. We use the transformation,
$f(T,B,T_G,B_G)=-T+\mathcal{F}(T, B, T_G, B_G)$ in order to obtain the
deviation from the Teleparallel Equivalent of General Relativity (TEGR). Two
cosmological models pertaining to the functional form of $\mathcal{F}(T, B,
T_G, B_G)$ have been studied. The well motivated forms are: (i) $\mathcal{F}(T,
B, T_G, B_G) = f_{0} T^{m} B^{n}T_{G}^{k}$ and (ii) $\mathcal{F}(T, B, T_G,
B_G)=b_{0} B + g_{0} T_{G}^{k} $. The evolutionary phases of the Universe have
been identified through the detailed analysis of the critical points. Further,
with the eigenvalues and phase space diagrams, the stability and attractor
nature of the accelerating solution have been explored. The evolution plots
have been analyzed for the corresponding cosmology and compatibility with the
present observed value of standard density parameters have been shown.
