Asymptotic analysis of integro-differential systems with an unstable spectral value of the integral operator's kernel

A system of integro-differential equations with rapidly varying kernels, one of which has an unstable spectral value, is considered. An algorithm based on the method of normal forms is proposed for finding asymptotic solutions (of an arbitrary order). The contrast structures (internal transition layers) in solutions to the problem are investigated by analyzing the leading term of the asymptotic expansion. It is shown that the contrast structures are caused by the instability of the spectral value and by the presence of inhomogeneity. The role of the kernels of the integral operators in the formation of contrast structures is clarified.