Dissertation

Invariant manifolds and applications for functional differential equations of mixed type

Promotor: S.M. Verduyn Lunel

Author

Hermen Jan Hupkes

Date

12 June 2008

Links

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.

• advanced and retarded arguments
• center manifold
• finite dimensional reduction
• hopf bifurcation
• invariant manifold
• life-cycle model
• lin's method
• mixed type functional differential equation