Singularly perturbed integro-differential systems with contrast structures

An integro-differential system with an eigenvalue of the limiting operator of the differential part taking the value 0 is considered. An algorithm is developed allowing one to obtain asymptotic solutions (of an arbitrary order) by the method of normal forms. Contrast structures (internal transition layers) in solutions of the problem under consideration are investigated on the basis of the analysis of the leading term of the asymptotic solution. Contrast structures are shown to result from the instability of the spectrum of the limiting operator and the presence of an inhomogeneity. The role of the kernel of the integral operator in the development of contrast structures is also cleared up. In integral systems with diagonal degeneration of the kernel $(K(t,t)\equiv0)$ the integral term plays no role in the development of contrast structures and, conversely, if the kernel is non-degenerate, then it is significant for the development of contrast structures.