On the fixed points and invariants of Dykhne transformations and the stability of the solutions of the problem of the effective conductivity of randomly inhomogeneous two-phase media

The fixed points and invariants of Dykhne transformations are determined. It is established that they correspond to the exact solutions and duality relations for the effective characteristics of an inhomogeneous medium. The stability of the exact solutions for the effective conductivity, which are fixed points of the Dykhne transformations, is studied by bifurcation-theory methods, and a classification of these fixed points by stability type is given. It is shown that the effective conductivity tensor of a two-phase medium in magnetic and ac electric fields for certain parameters of the medium can be an unstable point of the “saddle” type.