In this thesis we construct a perturbation method for delay differential equations (DDEs) based on the method of multiple scales for ordinary differential equations (ODEs) and ordinary difference equations (O$\Delta$Es). The method works for nonlinear DDEs, which are linear DDEs in the unperturbed case. The validity of the method is proven under certain conditions, such as a Lipschitz condition on the perturbation, and we illustrate how the method can be applied by working out several examples. We consider a delayed version of Mathieu's equation, which is especially useful, because it can be used when one linearizes a nonlinear oscillator around a period soluction. We also consider a quadratic perturbation. For these examples we have to analyse the relationship between the solutions of the characteristic equation. There already exists a perturbation method for DDEs, for which one solves a corresponding ODE, and uses this solution as an approximation. This method is only applicable when the influence of the delay is small, and is not always accurate due to the different natures of DDEs and ODEs. We study an example for which this method can be used, and show when it fails to give an accurate approximation. We then show how to use our perturbation method for this example, to obtain an accurate approximation.

In the thesis of N. Batenburg unexpected domain boundary behaviour was observed for chlorine on a copper surface. The aim of this thesis is to create a model for the behaviour of adsorbed chlorine atoms on a copper surface, to better understand the domain boundary behaviour. The copper was cut along the (111) Miller plane, resulting in a hexagonal lattice. We present the Ising and the hard hexagon model before moving onto the full model used for the chlorine atoms on the copper surface.

For the final model first only two-point interaction was considered. It was found that the energy density of the domain walls found by N. Batenburg was higher than the energy density of other types of domain walls. Only when extreme values for the parameters are used, the energy density of the domain walls found by N. Batenburg become the lowest. However, when these parameters are used in a simulation, other congurations, which were not considered in the analysis, are found. We conclude that solely considering two-point interaction results in an inaccurate model, as the domain boundary behaviour found by N. Batenburg cannot be reproduced. An attempt was made to include three-point interaction,
but due to a lack of time this attempt has not been successful. The three-point interaction has to be investigated further to create a more accurate model. Furthermore, the topology of the system could prove to be key to creating an accurate model.