The notes cover some main ideas in the theory of G convergence for ordinary differential equations, discussing the problem of non uniqueness of solutions to Cauchy problem (Peano phaenomenon). Homogeneization is discussed both in the linear and in the non-linear case; the chessboard homogeneization first introduced by the author is also presented, and seems to be a striking example of a non regular mean, namely it is possible to change monotonically the coefficients on a non zero set, without affecting the final mean.
The paper ends with a short presentation of De Giorgi's Gamma convergence in its full casisistic with a critical example that separates the four types of Gamma ++, Gamma +-, Gamma -+ and Gamma -- G. convergence. This is clearly the fundamental choice to perfom when applying these theories to calculus of variations.
