Regularization of neutral delay differential equations with several delays
Nicola Guglielmi and Ernst Hairer
Abstract.
For neutral delay differential equations the right-hand side
can be multi-valued, when one or several delayed arguments cross a
breaking point. This article studies a regularization via a singularly
perturbed problem, which smooths the vector field and removes the
discontinuities in the derivative of the solution. A low-dimensional
dynamical system is presented, which characterizes the kind
of generalized solution that is approximated. For the case
that the solution of the
regularized problem has high frequency oscillations around
a codimension-2 weak solution of the original problem,
a new stabilizing regularization is proposed and analyzed.
Key Words.
Neutral delay differential equations, generalized solutions,
singularly perturbed delay differential equations, stabilized regularization.