Graphs and asymptotic expansions in homogenisation of nonlinear equations

In order to pinpoint technical difficulties for the open problem of homogenisation of semilinear transport equation, following Tartar`s suggestion we study a nonlinear ordinary differential equation. We shall describe a method for computing the corenspoding asymptotic expansion. Due to their complexity, we represent the multiple integrals as graphs, which allows us to work with a number of leading terms, in a manner which might bear some similarity with Feymann diagrams.