Asymptotic Expansions for Regularized State-Dependent Neutral Delay Equations

Abstract

Abstract. Singularly perturbed delay differential equations arising from the regularization of state dependent neutral delay equations are considered. Asymptotic expansions of their solutions are constructed and their limit for ε → 0 + is studied. Due to discontinuities in the derivative of the solution of the neutral delay equation and the presence of different time scales when crossing breaking points, new difficulties have to be managed. A two-dimensional dynamical system is presented which characterizes whether classical or weak solutions are approximated by the regularized problem. A new type of expansion (in powers of √ ε) turns out to be necessary for the study of the transition from weak to classical solutions. The techniques of this article can also be applied to the study of general singularly perturbed delay equations.

Regularization of Neutral Delay Differential Equations with Several Delays

“…Reversing time, the same reasoning of [7] shows that in the "from above" case the solution of (16) behaves like…”

“…It is therefore of interest to study if, for ε → 0, the solution of (11) approximates a solution of (1) and, in the case of non-uniqueness of the solution of (1), which of the solutions will be approximated by (11). These questions have been addressed in [7] for the case of one delay and for codimension-1 weak solutions. Our aim is to get insight into the more complicated case of codimension-2 weak solutions.…”

“…The first situation has been studied in [7] and we briefly present the results that are important for the present work. We then give a formal derivation of the possible behaviors in the second situation without giving rigorous error estimates.…”

“…The initial condition η(0) = 0 follows from (13), and ζ(0) = α 1 (y * )z * = g(1) > 0 is a consequence of the assumption that the solution of (11) enters transversally the manifold α 1 (y) = 0, i.e., Up to now we do not know whether the smooth part of the solution (14) corresponds to a classical or to a weak solution. One of the main results of [7] tells us that one of the following two situations occurs:…”

“…In Section 3 we discuss a regularization via a singularly perturbed delay differential equation. We recall some results of [7] that concern the codimension-1 case, and we extend them to the codimension-2 situation. In particular we present a 4-dimensional dynamical system for which, near a breaking point in M 1 ∩ M 2 , the stationary points characterize the kind of solution (classical or weak) that is approximated by the regularization.…”