Regularization of neutral delay differential equations with several delays (Q1949247)

This paper is concerned with the system of neutral delay differential equations \[ \dot{y}(t)=f(y(t), \dot{y}(\alpha_1(y(t))), \cdots, \dot{y}(\alpha_m(y(t))) \text{ for } t>0, \] \[ y(t)=\varphi(t) \text{ for } t\leq 0, \] with smooth functions \(f(y, z_1, z_2, \cdots, z_m)\), \(\varphi(t)\) and \(\alpha_j(t)\). The authors consider time intervals where the solution satisfies \(\alpha_j(y(t))<t\) for all \(j\). First, a rigorous definition of generalized (classical and weak) solutions relating them to differential-algebraic systems of index 2 is given. Then the authors discuss a regularization via a singularly perturbed delay differential equation. They extend their earlier results concerning the codimension-1 case to the codimension-2 situation. In particular, they present a 4-dimensional dynamical system for which, near a breaking point in the codimension-2 manifold \(\{y: \alpha_1(y)=0\} \cap\{y: \alpha_2(y)=0\}\), the stationary points characterize the kind of solution (classical or weak) that is approximated by the regularization. A stabilizing regularization is proposed which, in many situations, eliminates the high oscillations. Finally, numerical experiments are given to illustrate the theoretical investigations.