SINGULAR PERTURBATION ANALYSIS OF BOUNDARY-VALUE-PROBLEMS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS .4. A NONLINEAR EXAMPLE WITH LAYER BEHAVIOR

A study is made of boundary value problems for a class of singularly perturbed nonlinear, second-order, differential-difference equations, i.e., where the highest-order derivative is multiplied by a small parameter. Depending on the region of parameter space, solutions of the nonlinear problem may not be unique, can exhibit extreme sensitivity to the values of the parameters, or may not exist. Typically, solutions exhibit layer behavior and/or exponentially large amplitudes. Approximate solutions of these boundary value problems are obtained by using singular perturbation methods and numerical computations and are then compared. Numerical computations of representative solutions illustrate the wide variety of possible behaviors.