Universiteit Leiden

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Invariant manifolds and applications for functional differential equations of mixed type

Promotor: S.M. Verduyn Lunel

Hermen Jan Hupkes
12 juni 2008
Thesis in Leiden Repository

Differential equations posed on discrete lattices have by now become a popular modelling tool used in a wide variety of scientific disciplines. Such equations allow the inclusion of non-local interactions into models and lead to interesting dynamical and pattern-forming behaviour. Although many numerical results have already been obtained for such lattice differential equations (LDEs), we are still far removed from a rigorous mathematical theory that is able to confirm and predict many of the interesting phenomena that these studies have uncovered. It is a basic and well-established mathematical practice to start the investigation of LDEs by looking for travelling waves. These are patterns that have a fixed shape and travel through the lattice at a fixed speed. Even this simple scenario however poses a significant challenge, due to the fact that a type of equation is encountered that defies treatment using classical techniques. This thesis describes several contributions towards the development of a new mathematical framework that will help to meet this challenge. The application range of the results is illustrated by discussing several problems encountered in various fields of research.

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