Pamplona, 23 January 2012 - 27 January 2012



This is the 9th winter school in Dynamical Systems of the DANCE (Dinámica, Atractores y Nolinealidad: Caos y Estabilidad) Spanish network. Download the Poster of the winter school.


mcin logo upna





Lluís Alsedà (Universitat Autònoma de Barcelona)
Enrique Ponce (Universidad de Sevilla)



Florentino Borondo (Universidad Autónoma de Madrid)
Alain Chenciner (Université Denis Diderot Paris 7)
Freddy Dumortier (Hasselt University)
Àngel Jorba (Universitat de Barcelona)
Patricia Yanguas (Universidad Pública de Navarra)



Manuel Iñarrea (Universidad de La Rioja)
Víctor Lanchares (Universidad de La Rioja)
Jesús Palacián (Universidad Pública de Navarra)
nAna Isabel Pascual (Universidad de La Rioja)
José Pablo Salas (Universidad de La Rioja)
Flora Sayas (Universidad Pública de Navarra)
Patricia Yanguas (Universidad Pública de Navarra)

Download as a PDF file


Scientific Sessions

 Coupled Systems of Differential Equations - Martin Golubitsky (Ohio State University)

Outline and Description

Coupled systems can be identified with directed graphs where each node represents a system of differential equations and each arrow represents coupling from one node to a second. A strong but often explored property of a solution to a coupled system is synchrony (where the output from two nodes is identical or equal after a phase shift). We explore how network architecture (the graph) affects the expected dynamics of the coupled system.  For example, what are the expected patterns of synchrony in coupled systems and what are the expected kinds of synchrony-breaking bifurcations.  Network symmetry motivates many of the questions that we ask, but symmetry is only part of the answer.
Topics to be covered
  1. Synchrony subspaces, balanced colorings, and quotient networks.
  2. Classification of regular networks with a small number of identical nodes.
  3. Synchrony-breaking bifurcations.
  4. Synchrony, phase-shift synchrony and multirhythms in periodic solutions.
  5. Feed forward networks.


  • Overview:  M. Golubitsky and I. Stewart. Nonlinear dynamics of networks: the groupoid formalism. Bull. Amer. Math. Soc. 43 No. 3 (2006) 305–364.
  • Technical:  M. Golubitsky, I. Stewart, and A. Torok. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dynam. Sys. 4(1) (2005) 78-100.
 Local Bifurcations and Reduction Methods in Reversible Systems. Application to Water Waves and Lattices - Gérard Iooss (Université de Nice)

Outline and Description

  1. Normal form theory. Application to elementary bifurcations of reversible systems in small dimensions.
  2. Center manifold reduction for infinite dimensional systems. Case of analytical vector fields. Analytic center manifold up to exponentially small term.
  3. Case of infinite dimensional reversible systems. Computation of coefficients of the normal form.
  4. Travelling waves in the water wave problem. Spatial dynamics formulation. Spectrum. Bifurcations of various types.
  5. Travelling waves in infinite lattices. Spatial dynamics formulation. Spectrum. Bifurcations.
  6. Remarks on "limiting" cases, where the reduction method does not apply. 


The following book can be used as a textbook for this course:

M.Haragus, G.Iooss: Local bifurcations, center manifolds, and normal forms in infinite dimensional systems. (EDP Sci. - Springer Verlag UTX series 2011 (329p.)). Chapters 2, 3 and 4 contain nearly all the lectures.
 Astrodynamics: Orbital Motion of Spacecraft in Strongly Perturbed Environments - Daniel Scheeres (University of Colorado)

Outline and Description

  1. Specific focus: Orbital Mechanics of spacecraft about asteroids, comets and planetary satellites. These environments are the most strongly perturbed environments found in nature for artificial satellites. Thus, they provide an excellent forum for the derivation and discussion of spacecraft orbital mechanics in general.
  2. Solved Problems of Astrodynamics: the 2-body problem and torque-free rotational motion. At the heart of astrodynamics are the non-trivial integrable problems of orbital motion of two point masses and the rotation of a torque-free rigid body. These solutions also serve as fundamental starting points for the more detailed description and discussion of non-integrable problems in this field.
  3. Models of the force environment at small bodies. To adequately discuss the dynamics of spacecraft several specific topics must be discussed, covering aspects of mass distribution models, gravitational potentials, radiation forces, and the like.
  4. The equations of motion: Lagrange's, Hamilton's, and Perturbation forms. Fundamentally, all motion in astrodynamics problems can be described using Newton's 2nd Law. However, there are many aspects of these problems that are hidden with such a fundamental approach, thus it is instructive to develop several different approaches to the mathematical description of these problems. A fundamental result is that (nearly) all astrodynamics problems can be reformulated into a Hamiltonian Dynamics form.
  5. Solutions and their properties: equilibria, periodic orbits, and general trajectories. Given that all astrodynamics problems can be reformulated into a Hamiltonian structure, there are strong constraints which can be placed on the properties of motion and solutions in these problems. This section provides detailed results on the stability of solutions and other fundamental constraints that exist for these problems.
  6. Computation of solutions and constraints: numerical, analytical and topological. Practical issues for the solution of astrodynamics problems are discussed. These include numerical integration, analytical theories of motion, and approximate theories based on veraging.
  7. Case studies: Applications of theory and detailed results for orbital motion in extreme environments. To provide insight into the actual implementation of astrodynamics theory, several different canonical problems will be investigated, motivated by actual systems found in the solar system.
    1. Uniformly rotating asteroids: 433 Eros
    2. Tumbling asteroids: 4179 Toutatis
    3. Small asteroids: 25143 Itokawa
    4. d) Binary asteroids: 1999 KW4
    5. Planetary satellites: The Hill 3-Body Problem


Any book on Celestial Mechanics would be a good complement to the lectures. Advanced texts include:

  • J.M.A. Danby," Fundamentals of Celestial Mechanics"
  • A.E. Roy, "Orbital Motion"
  • V. Szebehely, "Theory of Orbits"

For a more general introduction to astrodynamics, consider the texts:

  • Bate, Mueller and White, "Fundamentals of Astrodynamics"
  • Prussing and Conway, "Orbital Mechanics"

A particularly insightful and concise review of Celestial Mechanics is provided in:

  • "Mathematical Aspects of Classical and Celestial Mechanics", V.I. Arnold, V.V. Kozlov and A.I. Neishtadt.

I will have proof chapters of my forthcoming book, "Orbital Motion in Strongly Perturbed Environments" to distribute to the class.

 Lluís Alsedà, Universitat Autonòma de Barcelona (Spain)
 Martin Himmel, University of Mainz (Germany)
 Primitivo Belén Acosta-Humánez, Universidad del Norte (Colombia)
 J. Tomás Lázaro, Universitat Politècnica de Catalunya (Spain)
 Enrique Ponce, Universidad de Sevilla (Spain)
 Ashraf Hamdy Owis, Cairo University (Egypt)
 Alexander Gutierrez Gutierrez, Universidad de Granada (Spain)
 Naeem  Alkoumi, Birzeit University (Palestine)
 Yebdri Mustapha, University of Tlemcen (Algeria)
 Carmen Núñez, Universidad de Valladolid (Spain)
 Óscar Eduardo Martínez, Universidad Sergio Arboleda (Colombia)
 Enrique Vigil Álvarez, Universidad de Oviedo (Spain)
 Héctor Castejón Díaz, Universitat Politècnica de Catalunya (Spain)
 Pablo Roldán, Universitat Politècnica de Catalunya ( Spain)
 Alejandro Luque, Universitat Politècnica de Catalunya (Spain)
 Víctor Lanchares, Universidad de La Rioja (Spain)
 Alexey Osipov, Free University Berlin (Germany)
 Jeremy Grant, Universitat Politècnica de Catalunya (Spain)
 Eustache  Muteba Ayumba, Independent Researcher (DR Congo)
 Benedetta Bizzarri, Universitat Politècnica de Catalunya (Spain)
 Ariadna Farrés Basiana, Observatoire de Paris (France )
 Lei Zhao, Observatoire de Paris et Université Paris 7 (France)
 André Vanderbauwhede, Ghent University (Belgium)
 Àngel Jorba, Universitat de Barcelona (Spain)
 Noemi Bozek, AGH University of Science and Technology (Poland)
 Olga Romaskevich, Moscow State University (Russia)
 Clementa Alonso, Universidad de Alicante (Spain)
 Anna Markova, Lobachevsky State University of Nizhni Novgorod (Russia)
 Renato Huzak, Hasselt University (Belgium)
 Josep-Maria Mondelo, Universitat Autònoma de Barcelona (Spain)
 Tareq  Amro , Universidad de Granada  (Spain)
 David Blázquez Sanz, Universidad Sergio Arboleda (Colombia)
 Cinta Domínguez Moreno, Universidad de Huelva (Spain)
 Isabel Checa Camacho, Universidad de Huelva (Spain)
 Francisco Javier Molero, Universidad de Murcia (Spain)
 Xian Liao, Université Paris-Est Créteil (France)
 Hani Mohammed, Cairo University (Egypt)
 Ivan Ovsyannikov, Nizhny Novgorod State University (Russia)
 Anna Tamarit Sariol, Universitat Politècnica de Catalunya (Spain)
 Patricia Sánchez Martín, Universitat Politècnica de Catalunya (Spain)
 Elísabet Vela Felardo, Universidad de Sevilla (Spain)
 Ekaterina  Kutafina, Hasselt University (Belgium)
 Roberta Fabbri, University of Firenze (Italy)
 Marta Canadell Cano, Universitat de Barcelona (Spain)
 Jasson Vanegas Guzmán, Universidad Pública de Navarra (Spain)
 Ana Simic, Universidad de Navarra (Spain)
 Jorge Mozo Fernández, Universidad de Valladolid (Spain)
 Bhagawan Subedi, Pokhara University of Nepal (Nepal)
 Oriol Castejón, Universitat Politècnica de Catalunya (Spain)
 Adrià Simon, Universitat Politècnica de Catalunya (Spain)
 Catalina Vich Llompart, Universitat Politècnica de Catalunya (Spain)
 Hernán Neciosup Puican, Universidad de Valladolid (Spain)
 Luz Myriam Echeverry, Universidad Sergio Arboleda (Colombia)
 Tryasuchev Petr, Tomsk Polytechnic University (Russia)
 Víctor Muñoz Villarragut, Universidad de Valladolid (Spain)
 Alba Hierro Fabregat, Universitat Politècnica de Catalunya (Spain)
 Oswaldo José Larreal, Universidad del Zulia (Venezuela)
 Neus Llado Gambin, Universitat Politècnica de Catalunya (Spain)
 Roberto  Barrio, Universidad de Zaragoza (Spain)
 Abraham De la Rosa, Universitat Politècnica de Catalunya (Spain)
 Marcos Rodríguez, Centro Universitario de la Defensa (Spain)
 Reema Sbeih, Birzeit University (Palestine)
 Manuel Iñarrea, Universidad de La Rioja (Spain)
 Abimael Bengochea, Universidad Nacional Autónoma de México (Mexico)
 Frits Veerman, University of Leiden (The Netherlands)
 Eric Siero, University of Leiden (The Netherlands)
 María Begoña Melendo, Universidad de Zaragoza (Spain)
 Alexey Kazakov, Nizhny Novgorog State University (Russia)
 Vladimir Chigarev, Nizhny Novgorod State University (Russia)
 Adriana Buica, Universitatea Babes-Bolyai (Romania)
 Sergio Alejandro Carrillo, Universidad de Valladolid (Spain)
 David Romero i Sánchez, Universitat Autònoma de Barcelona (Spain)
 Juan Ramón Duque, Universidad Complutense de Madrid (Spain)
 Ekundayo Remilekun Oyekan, Ladoke Akintola University of Technology (Nigeria)
 Xavier Barrachina Civera, IUMPA - Universitat Politècnica de València (Spain)
 Flora Sayas, Universidad Pública de Navarra (Spain)
 Isaac García, Universitat de Lleida (Spain)
 Susanna Maza, Universitat de Lleida (Spain)
 Jordi Canela Sánchez, Universitat de Barcelona (Spain)
 Daniel Pérez Palau, Universitat de Barcelona (Spain)
 David Martí Pete, Universitat de Barcelona (Spain)
 Diego Fregolente Mendes de Oliveira, Max Planck Institute (Germany)
 Esther Barrabés Vera, Universitat de Girona (Spain)
 Narcís Miguel i Baños, Universitat de Barcelona (Spain)
 Josep M. Cors, Universitat Politècnica de Catalunya (Spain)
 Sergio Blanes, Universidad Politécnica de Valencia (Spain)
 Eva Tresaco Vidaller, Centro Universitario de la Defensa (Spain)
 Fernando Blesa, Universidad de Zaragoza (Spain)
 Joan Carles Tatjer, Universitat de Barcelona (Spain)
 Alan Marc Watson, University of Utah (United States)
 Zhaoyang Dong, Universitat Autònoma de Barcelona (Spain)
 Philipp Bader, Universidad Politécnica de Valencia (Spain)
 Ángeles Dena, Centro Universitario de la Defensa (Spain)
 Santiago Ibáñez, Universidad de Oviedo (Spain)
 Javier Ros Padilla, Universidad de Sevilla (Spain)
 Patricia Yanguas Sayas, Universidad Pública de Navarra (Spain)
 Jesús F. Palacián Subiela, Universidad Pública de Navarra (Spain)
 Sergey Lazarev, Lobachevsky State University of Nizhni Novgorod (Russia)
 Kate Sidorova, Lobachevsky State University of Nizhni Novgorod (Russia)
 Weimar  Muñoz , Universidad Sergio Arboleda (Colombia)
 Jorge Alberto Soliz Torrico, Universitat Politècnica de Catalunya (Spain)
 Smita Pati, Birla Institute of Technology (India)
 Shokry Nada, Qatar University (Qatar)
 Andrey Alpeev, Saint-Petersburg State University (Russia)
 Yeny Magaly Borda Pinzón, Universidad Sergio Arboleda (Colombia)
 Karen Rocío Pérez Silva, Universidad Sergio Arboleda (Colombia)
 Rafael Obaya, Universidad de Valladolid (Spain)
 Albert Granados, University of Stuttgart (Germany)
 Cristina López Godínez, Universitat Politècnica de Catalunya (Spain)
 Samson Folarin, Federal University of Technology (Nigeria)
 Farida Baidolda, Observatoire de Paris (France)
 Elisa Maria Alessi, University of Pisa (Italy)
 Davide Farnocchia, University of Pisa (Italy)
 Rosa M Herrera , APYCE (Spain)
 Antonio Algaba Durán, Universidad de Huelva (Spain)
 Manuel Reyes Columé, Universidad de Huelva (Spain)
 Cristóbal García García, Universidad de Huelva (Spain)
 Ana Isabel  Pascual Lería, Universidad de La Rioja (Spain)
 José Pablo Salas Ilarraza, Universidad de La Rioja (Spain)
 Sami Baraket, King Saud University (Saudi Arabia)
 Nejmeddine Chorfi, King Saud University (Saudi Arabia)
 Daniel Scheeres, University of Colorado (USA)
 Daniel Casanova Ortega, Universidad de Zaragoza (Spain)
 Martin Golubitsky, Ohio State University (USA)
 Gérard Iooss, Université de Nice (France)
 Jorge Galán Vioque, Universidad de Sevilla (Spain)