Advanced Course on
Taylor Methods and Computer Assisted Proofs - CAP08

To be held at the Institut de Matematiques de la Universitat de Barcelona from June 3 to 7, 2008, both included.


Below we present the contents of an Advanced Course mainly devoted to the Ph.D. students at the Facultat de Matemàtiques of the Universitat de Barcelona, and specially with interest in Dynamical Systems and related topics, but also with a wide applicability to other domains where similar techniques can be used.

The course is open to Ph.D. students, post-doctoral and senior researchers interested on these topics, aiming at an introduction to the key ideas and methodology or looking for fully detailed, complete and efficient implementations.

Effective computations done with computer have to take into account the effect of errors, unless they deal only with natural or rational numbers. One can think about problems in Computational Algebra, Computational Geometry, Symbolic Manipulation, Singularity Theory, Bifurcation Theory, Optimization, etc. Many of these aspects find a good domain of applicability in scientific, technical and industrial applications.

Going back to Dynamical Systems and related topics (ODE, DAE, PDE, DDE), computations far away from perturbative regimes can typically be done only using numerical methods. The problem is then how to prove rigorously existence of relevant objects, how many objects exist, stability properties, etc. Using tools from topology, analysis and geometry like the theories of operators, singularities, normal forms, transversality, etc, one tries to reduce the problem to a finite number of numerical checks (e.g., inequalities, inclusions). They are carried out using implementations of interval arithmetics, with algorithms that allow to prevent from a too large increase in the size of the intervals along the computations.

Prof. Martin Berz is a world leader in that domain. He has produced many papers and given lectures at different world conferences. See for a detailed account. Furthermore his collaborator Prof. Kyoko Makino (see will take care of technical details concerning examples and implementations. Other collaborators will also help with some concrete questions.

Scheme of the contents of the Advanced Course:

  1. Foundations of Rigorous Computing Interval methods, floating point requirements and standards, dependency problem, Taylor methods and related approaches, rigorous higher order bounds, paths to rigorous arbitrary precision and practical realization.
  2. Rigorous Integration of ODEs and Flows. Interval-based integration, error estimation, wrapping effect, differential algebraic structures, Taylor integration of flows, rigorous error bounds, automatic step size control.
  3. Constraint Satisfaction Taylor models for inverses, point solutions, constraint manifolds, differential algebraic equations (DAEs), rigorous high-order control theory.
  4. Divide and Conquer Methods. Taylor manifolds and automatic domain decomposition, applications for rigorous global optimization, constraint satisfaction, and flow integration.
  5. Applications in Dynamical Systems. Rigorous enclosures of attractors of discrete and continuous systems in two and higher dimensions, rigorous higher order normal forms, enclosures of hyperbolic manifolds in various dimensions, rigorous enclosure of homoclinic points and determination of symbolic dynamics, rigorous sharp estimates of topological entropy, center manifolds and nonlinear Lyapunov and pseudo-Lyapunov functions, rigorous Nekhoroshev-type long-term stability estimates.

For each of the topics, one shall work out example problems for the participants and provide in depth examples of the use of the methods.

Also, it is planned to use the various aspects of topic number 5) and combine them with the four previous topics, so that they become examples for each of the methods and do not have to wait until the very end.


How to arrive


To register for the course send a message to cap08 at The following information has to be included:

   Affiliation (Department, University, Country, etc):
   Domain of interest:
   e-mail, phone:

Deadline for inscriptions: May 12, 2008.


There is an inscription fee of 50 euros payable to the following place before the deadline.
Name of the bank: La Caixa
Account number: 2100-3642-12-2200093134
IBAN code: ES38 2100-3642-12-2200093134
When making the payment please mention explicitly CAP08 as concept. Send a copy of the order of transfer to IMUB, either by scanning it and sending the file by e-mail or by fax to the number at the bottom.


There will be a limited number of young researcher grants, to give some support for the stay. The concrete number and amount will depend on the availability of funds from different sources which have not yet been secured.

The persons asking for a grant should send a CV to the same address used to register, with the words "grant for CAP08" in the subject. They should ask for a support letter by a senior researcher, to be sent to the same address with subject "letter of support application of XXX" and including e-mail and phone of the supporter.


Participants in need of room for the stay should take care of this by themselves.

The file residencies.html provides a list of suggestions. The most convenient place, concerning distance to the Institut de Matematiques de la Universitat de Barcelona is the Residencia d'Investigadors, located at the Carrer de l'Hospital, 64, at 10 minutes walking from IMUB. Several other residences are very well communicated by underground.


Beyond the IMUB, the course is supported by the Facultat de Matemàtiques of the UB, the spanish Ministerio de Educación y Ciencia and the research teams on Dynamical Systems of the UB.

Institut de Matemàtica 
Gran Via de les Corts Catalanes 585,
08007 Barcelona, Spain
Phone: (+34) 93 402 13 85
Fax:  (+34) 93 403 59 63

Last modified: May 22, 2008