From June 20 to July 1, 2011 the IMA will offer an intensive
short course on modern mathematical tools for the study of
dynamical systems and their applications. The course is
designed for researchers in the mathematical sciences and
related disciplines. The main lecturers for the course are
Peter Bates, Department of Mathematics, Michigan State
University, and Rafael de la Llave, Department of Mathematics,
University of Texas at Austin.
Additional short introductions
to computational methods will be provided by Alex Haro
Provinciale, Departament de Matemàtica Aplicada i
Universitat de Barcelona, Spain, and Gemma
Huguet, Center for Neural Science, New York
University. Other guest lecturers include Martin
Lo, Jet Propulsion
Lab, Stephen Schecter, Department of Mathematics, North
Carolina State University, and George
Sell, School of Mathematics, University of Minnesota.
The primary audience for the course is mathematics faculty.
Some background in dynamical systems is expected. Participants
will receive full travel and lodging support during the
One way to understand complicated dynamical behavior is to find
landmarks or robust structures that organize the behavior.
Ideally one would like to find sets that are invariant under
the dynamics and on which the behavior is relatively simple.
Furthermore, it would be desirable if all dynamical behavior in
the system were somehow governed by the behavior on the
invariant sets, at least asymptotically in time.
Once one has found these (either numerically or analytically)
one can use them for other purposes, such as constructing
orbits with some desirable properties.
The most useful and best known invariant objects that are
persistent under perturbations are normally hyperbolic
invariant manifolds and quasiperiodic solutions.
The lecturers will present some recent theoretical
developments, numerical calculations and applications.
Lecture Series I: Quasiperiodic solutions. Lecturer: R. de la
Background in Analysis, number theory and symplectic geometry.
Kolmogorov's theorem on persistence of quasi-periodic motions.
Parameter dependence. Weakening the non-degeneracy conditions.
Variational KAM theory; Whiskered tori and their role in
Lecture Series II: Normally Hyperbolic Invariant
Lecturer: P. W. Bates.
Local stable, unstable, center-stable, center-unstable, and
center manifolds in Rn.
Extensions to semiflows in Banach spaces.
Applications to stability or instability.
Persistence of global NHIMs and foliations for semiflows under
From approximations to true NHIMs
Applications to singularly perturbed parabolic problems.
Short Lecture Series: Numerical Computations.
Haro and G. Huguet
- Manipulation of Polynomials, Fourier series, Fast Fourier
Numerical implementation of computations of invariant tori.
Validation of computations. Interval arithmetic in function
Additional guest lecturers will cover applications and other
aspects of computation.
The IMA New Directions Short Course
will be limited to 25 participants
selected by application. All
successful applicants will be funded for travel and local
expenses. Please see the IMA
policy for details