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 third of these will be take place on

Friday June 6, 2008,
at the Vrije University in Amsterdam
room S-209 of the Math (FEW) Building
De Boelelaan 1081

in walking distance from trainstation Amsterdam Zuid/WTC.
(Directions, see http://www.math.vu.nl/en/contact.php.)

Everybody is cordially invited.

Program

14:00-14:45 Olga Lukina (University of Groningen)
Global properties of integrable Hamiltonian systems

15:00-15:45 Bob Rink (Vrije Universiteit Amsterdam)
Rigorous integrable continuum approximations for the evolution of long waves in Fermi-Pasta-Ulam chain

16:00-16:45 Jean-Phillipe Lessard (Rutgers University and Vrije Universiteit)
Topological methods and rigorous numerics for the study of pattern formation models

Drinks afterwards

Abstracts

Olga Lukina

In the talk we will be interested in symplectic torus bundles that occur in integrable Hamiltonian systems, which we call the Lagrangian bundles. We review the theory of obstructions to triviality, which was developed by H.Duistermaat in 1980, in particular, monodromy, as well as the ensuing classification problems which involve the Chern and the Lagrange class. We elucidate some aspects of the theory. In particular, we stress the role of the integer affine structure, and explain in detail the symplectic classification of Lagrangian bundles with fixed integer affine structure.

Bob Rink

The Fermi-Pasta-Ulam experiment is famous for the unexpected recurrent behavior of long waves. Attempts to explain this observation usually involve an approximation by a KdV equation and a hand-wavy KAM argument. A rigorous justification for the KdV approximation was given only very recently though. Such finite-time results were obtained independently by Bambusi-Ponno and Wayne-Schneider. In this talk I will show how to derive the KdV equation as a resonant normal form, and I will present an improvement to this first approximation.

Jean-Phillipe Lessard

The theoretical starting point in the study of spatiotemporal pattern formation in systems is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. For potential models, it can be argued that the dynamics will eventually cease and thus reach an equilibrium state. Understanding the steady state solutions becomes a natural problem. In this talk, we propose a rigorous numerical method for the study of equilibrium solutions of a pattern formation model, namely the classical Swift-Hohenberg PDE defined on a rectangular spacial domain. The idea is to think of an equilibrium of the PDE as a fixed point of a mapping F and to prove that F is a contraction in a specific Banach space. The Banach fixed point theorem will then yield the existence of an equilibrium solution of the model. This is joint work with Marcio Gameiro.

The seminar organizers are Heinz Hanßmann and Rob Vandervorst.