Vrije Universiteit Amsterdam
Department of Mathematics

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Description of Bob Rink's research

My interests are in spatially extended dynamical systems such as lattices and partial differential equations. In particular I want to understand how symmetry, reversibility, coupled cell structure, variational character, Hamiltonian structure and other geometric properties affect the dynamics of these systems.

An example of such a dynamical system is the Fermi-Pasta-Ulam chain, for which I proved the applicability of KAM theory. I am also very interested in the continuum approximation of this well-known physical system.

I apply ideas from Aubry-Mather theory in the context of lattice models. Together with Blaz Mramor I proved the existence of a so-called "ghost circle" in this setting: a smooth interpolation of the Aubry-Mather set with a special dynamical meaning. Its existence implies that there are often many non-minimal equibrium states. Other than this, I am interested in the "destruction" of invariant circles and foliations. We proved that these geometric objects can be destroyed if their rotation number is easy to approximate by rational numbers.

More recently, I have studied coupled cell networks. Together with Jan Sanders and Eddie Nijholt, I was able to characterize these as equivariant dynamical systems. This has many implications for the dynamics and bifurcations of networks.