Seminars on Nonlinear Dynamics of Natural Systems

Similar to the Dynamics of Patterns Days we plan to regularly have (half)day mini-seminars on nonlinear dynamical systems, organized by and for the NDNS+ cluster but open to everyone interested. The second of these will take place on

Friday 8 December 2006, at the University of Utrecht room 611 of the Math Building Budapestlaan 6.

Directions: From Utrecht CS take bus 11 to `Botanische Tuinen', the closest bus station to the Math (Wiskunde) Building. For directions from there on see the map of the university.

Everybody is cordially invited.

Program

14:00-14:45 Jan Bouwe van den Berg (Vrije Universiteit Amsterdam) Closed characteristics on non-compact energy surfaces

15:00-15:45 Federico Camia (Vrije Universiteit Amsterdam) Random colorings, phase transitions and conformal invariance

16:00-16:45 Hicham Zmarrou (Universiteit van Amsterdam) Bifurcations of random circle diffeomorphisms

Abstracts

Jan Bouwe van den Berg

The study of periodic solutions of Hamiltonian equations has a long history, especially in the context of symplectic geometry, and a variety of results is known for compact energy levels (i.e. all motions are bounded a priori). In this talk we will consider Hamiltonians coming from classical mechanics and leading to non-compact energy manifolds. We use a suitable variational principle to prove that there must be at least one periodic motion on each energy level that satisfies certain topological and geometric conditions.

Federico Camia

Consider the random coloring of a regular hexagonal tiling of the plane in which each hexagon is colored blue or yellow with equal probability, independently of the other hexagons. Letting the mesh of the tiling tend to zero and focusing on the boundaries between blue and yellow regions reveals a complex geometric structure, with the boundaries converging to conformally invariant, random, fractal curves. In recent years, substantial progress has been made in understanding this limit with a combination of methods coming from discrete probability, the theory of stochastic processes and complex analysis. In turn, this has shed new light on the theory of (second order) phase transitions, allowing, among other things, the rigorous verification of many predictions made by physicists in the last thirty years. In this talk, I will describe some of this recent progress, focusing on work done in collaboration with C. M. Newman.

Hicham Zmarrou

We discuss iterates of random circle diffeomorphisms with identically distributed noise, where the noise is bounded and absolutely continuous. Adapting arguments of V.A. Kleptsyn and M.B. Nalskii, we prove a result on the occurrence of a unique random attracting fixed point. Bifurcations leading to an explosion of the support of a stationary measure from a union of intervals to the circle are treated. Here we show that this typically involves a transition from a unique attracting random periodic orbit to a unique attracting random fixed point. We briefly discuss higher dimensional generalizations.

The seminar organizers are Rob Vandervorst and Heinz Hanßmann.