Asymptotics of a step-like contrast structure with a multizonal interior layer

A boundary value problem for a singularly perturbed system of two second order ODEs with different degrees of the small parameter at the second derivatives is considered. The peculiarity of the problem is that one of the two equations of the degenerate system has three disjoint roots, two of which are double, and the third is simple (single). It is proved that for sufficiently small values of a small parameter, the problem has a solution with a fast transition from one double root of the degenerate equation to another double root in a neighborhood of some interior point of the segment. A complete asymptotic expansion of this solution is constructed and substantiated. It differs qualitatively from the well-known expansion in the case when all the roots of the degenerate equation are simple. In particular, the expansion is carried out not in integer but in fractional powers of the small parameter, the boundary layer variables have a different scale, and the transition layer turns out to be eight-zone.