Degree counting and Shadow system for Toda system of rank two: One bubbling

We initiate the program for computing the Leray–Schauder topological degree for Toda systems of rank two. This program still contains a lot of challenging problems for analysts. As the first step, we prove that if a sequence of solutions $(u_{1k},u_{2k})$ blows up, then one of $\frac{h_j{e}^{u_{jk}}}{{\int }_M{h}_j{e}^{u_{jk}}d{v}_g}$, $j=1,2$ tends to a sum of Dirac measures. This is so-called the phenomena of weak concentration. Our purposes in this article are (i) to introduce the shadow system due to the bubbling phenomena when one of parameters ${\rho }_i$ crosses 4π and ${\rho }_j?4\pi \mathbb{N}$ where $1\le i\ne j\le 2$; (ii) to show how to calculate the topological degree of Toda systems by computing the topological degree of the general shadow systems; (iii) to calculate the topological degree of the shadow system for one point blow up. We believe that the degree counting formula for the shadow system would be useful in other problems.