Logroño, 22 January 2018 - 26 January 2018



This is the 15th winter school in Dynamical Systems of the DANCE (Dinámica, Atractores y Nolinealidad: Caos y Estabilidad) Spanish network.

The School will take place at "Aula 104" of the Faculty of Science and Technology in Universidad de La Rioja.

A Poster session will be organized during the School for young participants. If you are interested please send title and abstract to the organizers before December 15, 2017.




Santiago Ibañez (Universidad de Oviedo)
Tere M. Seara (Universitat Politècnica de Catalunya)


Alessandra Celletti (University of Rome Tor Vergata)
Tibor Krisztin (University of Szeged)
Carmen Núñez (Universidad de Valladolid)
Roberto Barrio (Universidad de Zaragoza)
J. Ángel Rodríguez (Universidad de Oviedo)


Víctor Lanchares (Universidad de La Rioja)
Jesús Palacián (Universidad Pública de Navarra)
Roberto Barrio (Universidad de Zaragoza)
Ana Isabel Pascual (Universidad de La Rioja)
Manuel Iñarrea (Universidad de La Rioja)

Below you can download the program. Please, notice that the scientific program starts on Monday 22 at 9:00 and it will be closed on Friday 26, at 11:30.


Scientific Sessions

 Poster session - Young participants

1. Gronwall-Bellman estimates for the solutions of some linear integral inequalities with delay. Sebastián Buedo Fernández (Universidade de Santiago de Compostela)

2. Existence, uniqueness and computation of the solution in linear fractional integral equations of arbitrary real order with constant coefficients. Daniel Cao Labora (Universidade de Santiago de Compostela)

3. Global connections in a Lorenz-like system. A. Algaba, C. Domínguez-Moreno, M. Merino, A. J. Rodríguez-Luis (Universidad de Huelva)

4. On some global bifurcations regarding periodic orbits in piecewise-linear systems with hysteresis. Marina Esteban (Universidad de Sevilla)

5. Existence and localization of fixed points for compressions or expanssions of a cone by using star convex sets. Cristina Lois Prados (Universidade de Santiago de Compostela)

6. Resonance tongues in the linear Sitnikov equation. Mauricio Misquero Castro (Universidad de Granada)

7. Some results on impulsive differential equations. José Manuel Uzal (Universidade de Santiago de Compostela)

8- Phase-Locked States in the Oscillatory Regime of Neuronal Networks. Alberto Pérez Cervera (Universitat Politècnica de Catalunya)

 Network dynamics and bifurcations - Peter Ashwin (University of Exeter)

Outline and description

We discuss some of the mathematical theory for bifurcations of dynamical systems, with application to network dynamics. The presence of special structures in the form of network symmetries constrain but give a much richer set of possible generic bifurcations that the non-symmetric case. We introduce the tools of generic bifurcation theory for equivariant (symmetric) systems and apply this to examples in neural network dynamics where many of these bifurcations are associated with onset or loss of various types of synchrony.

Topics to be covered

1. Bifurcation theory for ODEs, bifurcations of equilibria and genericity

2. Centre manifolds, normal forms and symmetries

3. Symmetric (equivariant) dynamics: group representation and bifurcations

4. Network dynamical systems and applications of symmetric bifurcation theory

5. Some examples in oscillatory neural network dynamic 


P Ashwin, S Coombes, R Nicks. Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience, J. Math. Neurosci. 2016: available from https://mathematical-neuroscience.springeropen.com/articles/10.1186/s13408-015-0033-6

M Golubitsky, I Stewart. The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Birkhauser 2002: http://www.springer.com/gp/book/9783764366094

J Moehlis and E Knobloch. Equivariant dynamical systems – Scholarpedia: http://www.scholarpedia.org/article/Equivariant_dynamical_systems


handouts and notes of the course

Outline and description

Often people informally confuse random and chaotic motion. Yet, the first is probabilistic while the latter is deterministic. It follows that to understand in which precise sense a deterministic motion can be called random is a non trivial issue. In fact, our theoretical understanding of such question is limited to very simple models. Also, there exists some disagreement on the role of randomness in the study of dynamical systems. I will describe some classes of systems for which something rigorous can be said and discuss future perspective. 

More precisely, I will try to give some ideas on how to investigate the statistical properties and establish limit theorems (Central Limit Theorem, Large deviations, averaging, ….) for expanding maps, hyperbolic systems, partially hyperbolic systems and, possibly, systems with small noise. 


C. Liverani. Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa. Centro di Ricerca Matematica "Ennio De Giorgi" : Proceedings. Published by the Scuola Normale Superiore in Pisa (2004). 

Jacopo De Simoi, C. Liverani. The Martingale approach after Varadhan and Dolpogpyat. In "Hyperbolic Dynamics, Fluctuations and Large Deviations", Dolgopyat, Pesin, Pollicott, Stoyanov editors, Proceedings of Symposia in Pure Mathematics, 89, AMS (2015). 

Blumenthal, Alex; Xue, Jinxin; Young, Lai-Sang. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. of Math. (2) 185 (2017), no. 1, 285–310. 


Notes of the course

 Resonances: from stability to chaos - Anne Lemaitre (Université de Namur)

Outline and description

The course will first give an overview of the different types of resonances appearing in celestial mechanics with numerous examples in the Solar System. The basic fundamental models will be presented and analysed, as well as the way to isolate (by averaging) the long-term resonant dynamics. The Hamiltonian formalism will be used to build the differential equations, and consequently its properties will be reminded.

Then the different types of resonances will be presented, with specifications and examples taken in various regions.

The captures into resonance or the crossings of a resonance can be modelled in some cases with the adiabatic invariant, in presence of dissipative forces. Probabilities of capture can be deduced in very simple models or through huge and complex numerical integrations.

Resonances could be associated with stability or with chaos, especially when they cross each other, to create a real web. Numerically and analytically these situations could be described or simply quantified, using well-known tools as chaos indicators (MEGNO or FLI or frequency analysis).

The understanding of resonant phenomena often implies to push the integrations on much longer time scales. They are then associated to numerical techniques, trying to avoid the increase of the energy, as symplectic integrators.

Topics to be covered

1. Resonances : surveys

1.1 Resonances In our solar system: examples, effect, history, a first panorama

1.2 General principles: closeness of two frequencies, isolation of a resonant angle, calculation of the equilibria, their stability, the separatrices.

1.3 Definition, models, examples and  simulations for different resonances: mean motion, secular, secondary, Kozai-Lidov, spin-orbit, gravitational, etc.

2. Capture into resonance in presence of dissipation

2.1 Adiabatic invariant theory

2.2 Main dissipative contributions

2.3 Capture or escapes

2.4 Simulations and applications

3. Chaos

3.1 Web of resonances

3.2 Chaos: measurement

3.3 Megno, FLI, frequency analysis

4. Long term integrations

4.1 Symplectic integrators

4.2 Validity and stability

4.3 Comparisons


Henrard J., 1982, Capture into resonance - an extension of the use of adiabatic invariants, Celestial Mechanics, 27, p. 3-22.

Jancart S. and Lemaitre A., 2001, Dissipative forces and external resonances, Celestial Mechanics and Dynamical Astronomy, 81, p. 75-80.

D’Hoedt S. and Lemaitre A , 2004, The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model, Celestial Mechanics and Dynamical Astronomy, 89, p. 267-283.

Lemaitre A., 2010, Resonances: Models and Captures, Dynamics of Small Solar System Bodies and Exoplanets, Eds. Souchay and Dvorak, Lecture Notes in Physics, 790 , p. 1-62.

Verheylewegen, E. and Lemaitre A., 2014, The 3:1 mean motion resonance between Miranda and the inner Uranian satellites, Cressida and Desdemona, Celestial Mechanics and Dynamical Astronomy, 119, p. 283-299.

Sansottera M., Lhotka Ch. And Lemaitre A., 2015, Effective resonant stability of Mercury, Monthly Notices of the Royal Astronomical Society, 452, 4, p. 4145-4152.

 Eduardo Liz, Universidade de Vigo (Spain)
 Miriam Steinherr Zazo, University of Bremen (Germany)
 Santiago Ibáñez, Universidad de Oviedo (Spain)
 Patricia Yanguas Sayas, Universidad Pública de Navarra (Spain)
 Jesús F.  Palacián Subiela, Universidad Pública de Navarra (Spain)
 Artur Prugger, University of Bremen (Germany)
 Cristina Lois Prados, Universidade de Santiago de Compostela (Spain)
 Carmen Núñez Jiménez, Universidad de Valladolid (Spain)
 Siemer Lars, University of Bremen (Germany)
 Víctor Lanchares, Universidad de La Rioja (Spain)
 Jichen Yang, University of Bremen (Germany)
 José Manuel Uzal Couselo, Universidade de Santiago de Compostela (Spain)
 Daniel Cao Labora, Universidade de Santiago de Compostela (Spain)
 Sebastián Buedo Fernández, Universidade de Santiago de Compostela (Spain)
 Naeem Alkoumi, Hebron University (Palestine)
 Marina Esteban, Universidad de Sevilla (Spain)
 Mauricio Misquero Castro, Universidad de Granada (Spain)
 Enrique Ponce Núñez, Universidad de Sevilla (Spain)
 Wael Shaheen, Hebron University  (Palestine)
 Ramakanta Meher, S.V National Institute of Technology (India)
 Venktesh Venktesh, Universidade de Santiago de Compostela (Spain)
 Óscar del Río Rodríguez, Universitat Politècnica de Catalunya (Spain)
 Manuel Iñarrea Las Heras, Universidad de La Rioja (Spain)
 M. Angeles Martínez Carballo, Universidad de Zaragoza (Spain)
 Roberto Barrio, Universidad de Zaragoza (Spain)
 Marcos Rodriguez, Universidad de Zaragoza (Spain)
 Sergio Serrano, Universidad de Zaragoza (Spain)
 Alvaro  Lozano Rojo, Universidad de Zaragoza (Spain)
 Doron Elad, Technion, Israel Institute of Technology (Israel)
 Raid Amro, Aquds Open University (Palestine)
 Begoña Nicolás , Universitat de Barcelona (Spain)
 Rosana Rodríguez-López, Universidade de Santiago de Compostela (Spain)
 Trushitkumar Patel, Sardar Vallabhbhai National Institute of Technology (India)
 Yuliya Bakhanova, Lobachevsky State University (Russia)
 Abdelqader Halman, Hebron University  (Palestine)
 Ashraf Owis, Cairo University (Egypt)
 Francisco Javier  Ros Padilla, Universidad de Sevilla (Spain)
 Alberto Pérez Cervera, Universitat Politècnica de Catalunya (Spain)
 Daniel J. Pagano, Universidade Federal de Santa Catarina (Brazil)
 Víctor Ortega, Universidad de Granada (Spain)
 J. Tomás  Lázaro, Universitat Politècnica de Catalunya (Spain)
 Ana Isabel Pascual Lería, Universidad de La Rioja (Spain)
 Otávio Marçal Leandro Gomide, Universitat Politècnica de Catalunya (Spain)
 Antonio Pumariño Vázquez, Universidad de Oviedo (Spain)
 Fernando Fernández-Sánchez, Universidad de Sevilla (Spain)
 Guillermo Alonso Alvarez, Universitat Politècnica de Catalunya (Spain)
 Robert Cardona Aguilar, Universitat Politècnica de Catalunya (Spain)
 Albert Jiménez Ramos, Universitat Politècnica de Catalunya (Spain)
 Pol Deulofeu Matas, Universitat Politècnica de Catalunya (Spain)
 Kenens Karel, Hasselt University (Belgium)
 David Rojas, Universidad de Granada (Spain)
 Ekaterina Shiryaeva, Lobachevsky State University (Russia)
 Marc Jorba-Cuscó, Universitat de Barcelona (Spain)
 Matteo Franca, Università Politecnica delle Marche (Italy)
 Miguel Pereira Hernández, Universitat Politècnica de Catalunya (Spain)
 Jose Pablo Salas Ilarraza, Universidad de La Rioja (Spain)
 Lucía Pérez, Universidad de Oviedo (Spain)
 Juan Ramón Pacha Andújar, Universitat Politècnica de Catalunya (Spain)
 Angel Jorba, Universitat de Barcelona (Spain)
 Iacopo Paolo Longo, Universidad de Valladolid (Spain)
 Joan Carles Tatjer, Universitat de Barcelona (Spain)
 Javier Montes Maldonado, Universidad Politécnica de Madrid (Spain)
 Tere Martinez-Seara, Universitat Politècnica de Catalunya (Spain)
 Alex Haro, Universitat de Barcelona (Spain)
 Rodrigo Gonçalves Schaefer, Universitat Politècnica de Catalunya (Spain)
 Iván Sánchez Sánchez, Universitat Autonòma de Barcelona (Spain)
 Enrique Vigil, Universidad de Oviedo (Spain)
 Edward Lakin, Universitat politécnica de Catalunya  (Spain)
 Jorge Galán Vioque, Universidad de Sevilla (Spain)
 Rafel Prohens Sastre, Universitat de les Illes balears (Spain)
 Telmo Peixe, ISEG-Lisbon School of Economics & Management, Universidade de Lisboa (Portugal)
 Anne Lemaitre, Université de Namur (Belgium)
 Peter Ashwin, University of Exeter (United Kingdom)
 Carlangelo Liverani, Università degli Studi di Roma (Italy)
 Santiago Díaz Madrigal, Universidad de Sevilla (Spain)
 Mª de la Cinta Domínguez Moreno, Universidad de Huelva (Spain)
 Susanna Maza, Universitat de Lleida (Spain)
 Isaac García, Universitat de Lleida (Spain)
 Jone Uria Albizuri, Basque Center of Applied Mathematics (Spain)
 Serafim Rodrigues, Basque Centre for Applied Mathematics (Spain)
 Rafael Obaya, Universidad de Valladolid (Spain)