We consider a quasi-linear heat transmission problem for a composite material which fills the n-dimensional Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies a cavity of size, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. For small enough the problem is known to have a solution, i.e., a pair of functions which determine the temperature distribution in the two materials. Then we prove a limiting property and a local uniqueness result for families of solutions which converge as tends to 0.