`Finite and Infinite Dimensional Dynamical Systems'
Contains among other things:
a) Persistence of dynamical properties,
like existence of periodical solutions, invariante manifolds,
among other things by structural stability.
b) Structure preserving normal forms with applications in dissipative,
Hamiltonian, reversible settings, etc.
c) Kolmogorov Arnold Moser Theory of quasi-periodic invariant tori
d) Applications of Singularity Theory in a number of examples
Literature:
1. H.W. Broer, Notes on perturbation theory 1991, Erasmus ICP
Mathematics and Fundamental Applications}, Aristotle University
Thessaloniki , (1993), 44 p.
2. M.C. Ciocci, A. Litvak-Hinenzon and H.W. Broer,
Survey on dissipative KAM theory including quasi-periodic
bifurcation theory based on lectures by Henk Broer.
In J. Montaldi and T. Ratiu (eds.):
Peyresq Lectures on Geometric Mechanics and Symmetry,
LMS Lecture Notes Series 306 ,
Cambridge University Press.
Aim of the Seminar
In this seminar, a number of concrete problems will be presented, that are motivating for the MRI
master class. In particular, the seminar brings together finite and infinite dimensional dynamical
systems,as, quite often, low dimensional approximating models are used to understand infinite
dimensional systems. The topics include (geometric) bifurcation theory, variational methods, stability
theory, delay equations, populations dynamics.
In the first semester a number of lectures will be given by various lecturers, who are willing to
supervise a final project. This final project will be carried out in the second semester, but in
consultation with the supervisor, one can also start earlier.
The second part of the seminar will be used to invite speakers who will lecture about their own
research. It will illustrate the the use of ideas from dynamical system theory in practise.
Ergodic theory, a combination of geometric methods and analytic methods with a stochastic flavor, enables an understanding of structures in dynamics when a complete description is beyond reach. In this course we develop and apply these techniques. In the second half of the course renormalization techniques are treated. These explain universal geometric structures in dynamics, that is, scaling properties of the dynamics that exist independent of the precise formulas for the dynamical system.
Structure of the course:
-- Pesin Theory
-- Stable and unstable foliations and their problems, by a number of examples.
-- Transfer operators and decay of correlation
-- Ergodic Theory in 1D dynamics
-- Renormalization
Literature:
R. Mane, Ergodic Theory and Differentiable Dynamics Ergebnisse der Mathematik und ihrer Grenzgebiete 3 (8). Springer-Verlag 1987.
From January 18 - March 1 Mats Gyllenberg (Helsinki, F.C. Donders-professor at Utrecht University January-March 2006) will give a series of lectures on
`Dynamics of Physiologically Structured Populations'
Summary :
The time-evolution of a (physiologically) structured population can in a natural way be modelled as a nonlinear infinite dimensional dynamical system. In general, such systems are notoriously difficult to analyse. However, in population dynamics the nonlinearity corresponding to interaction between individuals is modelled, not as direct interaction, but as interaction through the environment. This makes the models more amenable to analysis.
The course starts with an account of the basic principles of modelling structured populations and will in the end bring the audience to the research frontier of the field of mathematical population theory. Topics include existence and uniqueness of solutions, steady-state analysis, stability and bifurcation theory, and adaptive dynamics.
Prerequisites for the course are a solid background in undergraduate mathematics (analysis, linear algebra, probability) and a genuine interest in population biology.
Variational methods are a central class of techniques in the study of Hamiltonian systems and differential equations. The use of functional analysis provides powerful tools for the analysis of a wide variety of problems, and in addition the variational viewpoint provides a separate, complementary view on issues in the field of differential equations. In this course we introduce and discuss such tools and apply them to various types of problems.
Structure of the course:
-- Preliminaries: convex functions, weak convergence
in Banach spaces, Sobolev spaces
-- Interpretation of solution of ODE or PDE as a stationary point of a functional
-- Weak solutions
-- Existence of minimizers of convex functionals, uniqueness
-- Proving additional regularity
-- Constrained minimization and the Mountain Pass lemma
-- Applications to nonlinear elliptic boundary-value problems
-- Hamiltonian structure of Maxwell's equations; transmittance
problems, wave guides, efficient boundary conditions
-- Resonant modes and states of defect grating structures as eigenvalue
problems
-- Hamiltonian formulation of Korteweg-de Vries and nonlinear Schroedinger equations
-- Coherent structures as Hamiltonian relative equilibria
Central to this course are the concepts of global attractor and inertial manifold of the flow generated by a partial differential equation (PDE). Through these concepts, the infinite dimensional dynamics of a PDE can be interpreted and studied as a finite dimensional dynamical system. The existence of a global attractor of finite (Hausdorff) dimension will be established for nonlinear reaction-diffusion equations. The global attractor is embedded in the inertial manifold, which is a finite dimensional manifold that attracts the (infinite dimensional) flow generated by the PDE.
Structure of the course:
-- The global attractor in a general setting: existence and properties
-- Weak solutions of nonlinear reaction-diffusion equations
-- The global attractor for reaction-diffusion equations
-- Finite dimensional attractors: theory and examples
-- Finite dimensional dynamics
-- The Navier-Stokes equations and/or the Kuramoto-Sivashinsky equation
Literature:
James C. Robinson, Infinite-Dimensional Dynamical Systems -- An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors , Cambridge Texts in Applied Mathematics.