Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Quations II. Rapid Oscillations and Resonances

Abstract

This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502–531). The previous paper focused on problems involving boundary and interior layer phenomena. Here the problems studied have solutions exhibiting rapid oscillations. The presence of the shift terms can induce large amplitudes, multiphase behavior, and resonance phenomena.n particular, we study two types of resonance phenomena, namely “global” and “local” resonance. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies. In a companion paper (SIAM J. Appl. Math., 45 (1985), pp. 708–734) we study problems with solutions which exhibit turning point behavior.