Intensive Courses

The courses will take place in room B1 of the "Facultat de Matemàtiques", in the historical building of the university. It is located on the ground level, in the Science Court. If you enter the historical building from the street "Gran Via" (where the main door is), the Science Court is at the right side. The only exception to this will be the course on Scientific Visualization, that will take place in the computer room IC (also on the ground level of the Science Court).

New warning: There is a small change of schedule in Week 4: The second session of Oct 18 (monday) and the first session of Oct 21 (thursday) are interchanged. That is, on Oct 18 the two sessions will be about Numerical integration of ODEs (II)  and on Oct 21 the two sessions will be about Computer assisted proofs in Dynamics.

Old warning: Due to the general strike in Spain in Sept 29th, the lectures on that day are moved to the evening of Sept 28th. The course on Numerical Integration of ODEs  will take place from 16:00 to 17:30 in room B1. The course on Scientific Visualization will take place from 18:00 to 19:30 in computer room IC.


 9:30 - 11:00
  Break
 11:30 - 13:00
Week 1:
27 sept - 1 oct
 Numerical Integration of ODEs

Scientific visualisation
Week 2:
4 oct - 8 oct
 Invariant manifolds
Zero finders, continuation and bifurcations
Week 3:
11 oct - 15 oct
 Free of Courses !!
Free of Courses !!
Week 4:
18 oct - 22 oct
 Numerical integration of ODEs II
Computer assisted proofs in dynamics
Week 5:
25 oct - 29 oct
 Normal forms and applications
Dynamics indicators


  1. Numerical integration of ODEs I : Geometric methods. Roberto Barrio (U. Zaragoza), Fernando Casas (U. Jaume I).

    Fundamentals for the numerical integration of ODEs. Stability, consistency and convergence. One step and multistep methods. Stiffness. Symplectic integration methods.

  2. Scientific visualization. Núria Pla (U. Politècnica de Catalunya), Anna Puig (U. Barcelona), Dani Tost (U. Politècnica de Catalunya).

    This course aims to introduce packages and libraries for visualizing scientific data with an emphasis on 3D visualizations and animations. The course covers different approaches in surface extraction, volume viewing, volume shading, color editing, available libraries, packages, and common data formats.

  3. Invariant manifolds. Alex Haro (U. Barcelona), Josep Maria Mondelo (U. Autònoma de Barcelona).

    Parameterizations of invariant manifolds and their functional equations. Introduction to symbolic computation of formal expansions and automatic differentiation. High order approximation of stable, unstable and centre manifolds of fixed points. Computation of invariant tori and their whiskers. Application to the dynamics in the neighborhood of the collinear libration points.

  4. Zero finders, continuation and bifurcations. Application to computation of periodic orbits. Jorge Galán (U. Sevilla), Francisco Javier Muñoz (U. Cardenal Herrera).

    Computation of zeros of maps, continuation with respect to parameters and bifurcations. Periodic orbits. Stability and bifurcations. Systems with additional first integrals. The Hamiltonian case.

  5. Numerical integration of ODEs II: Taylor methods and jet propagation. Extended precision. Applications. Àngel Jorba (U. Barcelona).

    Automatic differentiation. Taylor methods for non-stiff ODEs. Use of extended arithmetic. Jet propagation and high order variational equations. Applications to Celestial Mechanics and Astrodynamics.

  6. Computer assisted proofs in dynamics. Warwick Tucker (U. of Bergen).

    Interval arithmetic. Zeros of functions. Krawczyk operator. Existence of fixed points. Validated computation of normal forms. Validated numerical integration of ODE. Applications: Lorenz problem, abundance of sinks, hyperbolicity estimates.

  7. Normal forms and applications. Carles Simó (U. Barcelona).

    Effective computation of normal forms for flows and maps. Use of ad-hoc algebraic manipulators: storage, retrieval and basic operations. Bifurcation analysis. Computation of several kinds of invariant objects. Detection of analytic/Gevrey character. From local to global. Combining the methods with jet transport.

  8. Dynamics indicators: Lyapunov exponents, frequency analysis, entropy and dimensions. Renato Vitolo (U. of Exeter),  Josep Maria Mondelo (U. Autònoma de Barcelona).

    Computation of Lyapunov exponents. Basic methods and variants. Fast Lyapunov indicators. Metric entropy computations. Extreme value laws for dynamical systems. Computation of dimenssions of attractors. Indicators on chaotic attractors. Examples for flows and discrete maps.