Asymptotic expansion of periodic solutions for a singular perturbation problem including nonlinear dynamical system with two boundary layers (Q1884513)

A singularly perturbed problem in the form of two first-order nonlinear differential equations, with periodic boundary conditions, one including a small parameter, on the basic interval \((a, w)\) is considered. The authors are looking for a periodic solution as a sum of an outer and two inner solutions in the form of asymptotic expansions. These asymptotic expansions are obtained for both outer and inner solutions by solving three systems of \(N+ 1\) algebraic and three systems of \(N+ 1\) first-order differential equations, where \(N\) is the degree of truncated asymptotic series. The unknown constants in these solutions are calculated from the periodic and matching conditions which give a system of \(6(N+ 1)\) equations that have to be solved step by step. Remark of the reviewer: The language used hinders the understanding.