  Here we initiate the program for computing the Leray-Schauder topological
degree for $SU(3)$ Toda system. This program still contains a lot of
challenging problems for analysts. The first step of our approach is to answer
whether concentration phenomena holds or not. In this paper, we prove the
concentration phenomena holds while $\rho_1$ crosses $4\pi$, and $\rho_2\notin
4\pi\mathbb{N}$. However, for $\rho_1\geq 8\pi$, the question whether
concentration holds or not still remains open up to now. The second step is to
study the corresponding shadow system and its degree counting formula. The last
step is to construct bubbling solution of $SU(3)$ Toda system via a
non-degenerate solution of the shadow system. Using this construction, we
succeed to calculate the degree for $\rho_1\in(0,4\pi)\cup(4\pi,8\pi)$ and
$\rho_2\notin 4\pi\mathbb{N}$.
