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IMA New Directions Short Course
Invariant Objects in Dynamical Systems and their Applications
June 20 - July 1, 2011

Peter BatesMichigan State University
Rafael de la LlaveThe University of Texas at Austin

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 Anàlisi, 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 workshop.

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 Llave.

  • 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 diffusion.


Lecture Series II: Normally Hyperbolic Invariant Manifolds. 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 perturbation.
  • From approximations to true NHIMs
  • Applications to singularly perturbed parabolic problems.

Short Lecture Series: Numerical Computations. Lecturers: A. Haro and G. Huguet

  • Manipulation of Polynomials, Fourier series, Fast Fourier Transform.
  • Numerical implementation of computations of invariant tori.
  • Validation of computations. Interval arithmetic in function space.

Additional guest lecturers will cover applications and other aspects of computation.


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