Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.

A multi-symplectic system is a PDE with a Hamiltonian structure in both temporal and spatial variables. This article considers spatially periodic perturbations of symmetric multi-symplectic systems. Due to their structure, unperturbed multi-symplectic systems often have families of solitary waves or front solutions, which together with the additional symmetries lead to large invariant manifolds. Periodic perturbations break the translational symmetry in space and might break some of the other symmetries as well. In this article, periodic perturbations of a translation invariant PDE with a one-dimensional symmetry group are considered. It is assumed that the unperturbed PDE has a three-dimensional invariant manifold associated with a solitary wave or front connection of multi-symplectic relative equilibria. Using the momentum associated with the symmetry group, sufficient conditions for the persistence of invariant manifolds and their transversal intersection are derived. In the equivariant case, invariance of the momentum under the perturbation gives the persistence of the full three-dimensional manifold. In this case, there is also a weaker condition for the persistence of a two-dimensional submanifold with a selected value of the momentum. In the non-equivariant case, the condition leads to the persistence of a one-dimensional submanifold with a seleceted value of the momentum and a selected action of the symmetry group. These results are applicable to general Hamiltonian systems with double zero eigenvalue in the linearization due to continuous symmetry. The conditions are illustrated on the example of the defocussing non-linear Schroumldinger equations with perturbations which illustrate the three cases. The perturbations are: an equivariant Hamiltonian perturbation which keeps the momentum level sets invariant; an equivariant damped, driven perturbation; and a perturbation which breaks the rotational symmetry.