Carmona, 26 January 2009 - 30 January 2009

PRESENTATION

Carmona



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

SPONSORS

Ministerio de Eduación y Ciencia
IMUB
logoConsolider
imus
US
logoUPO

COMMITTEES

COORDINATORS

Àngel Jorba (Universitat de Barcelona)
Carmen Núñez (Universidad de Valladolid)

SCIENTIFIC COMMITTEE

Amadeu Delshams (Universitat Politècnica de Catalunya
Jorge Galan (Universidad de Sevilla)
Santiago Ibañez (Universidad de Oviedo)
Rafael Obaya (Universidad de Valladolid)

ORGANIZING COMMITTEE

Jorge Galan (US)
Javier Ros Padilla (US)
Enrique Ponce (US)
Fernando Fernadez (US)
Francisco Torres (US)

Download as a PDF file

Download
 Slow-fast systems and relaxation oscillations - Freddy Dumortier (Hasselt University)

Abstract

In these lectures we want to study systems depending on multiple time scales and present geometric techniques to study such systems (by “geometric” we mean coming from “regular” dynamical systems theory). In fact we will focus on two different time scales, hence on so-called “slow-fast”systems. We will essentially limit to two-dimensional systems, however depending on an arbitrary number of parameters. Our attention will primarely go to periodic orbits, called relaxation oscillations, and the bifurcations that they undergo. Special attention will be given to Liénard equations.

Some points of attention:

  • center manifolds near normally hyperbolic singular curves
  • role of slow divergence integral
  • canard solutions
  • (family) blow up of contact points
  • generic breaking mechanisms and entry-exit relations
  • multi-layer canard cycles and translated power functions
  • slow dynamics with singularities
  • birth of canard cycles
  • Hilbert’s 16th problem, Smale’s 13th problem and Roussarie’s program
  • compactification of spaces of polynomial Liénard equations

 

Course notes will be distributed. The notes are based on different papers about singular perturbations by Robert Roussarie, Peter De Maesschalck and myself , that have already been published (see e.g. MathSciNet) or that will be made available during the course week. Prerequisites can be found in the chapters 1,2,3,5 and 7 of the book:
F.Dumortier, J.Llibre, J.C.Artés: “Qualitative theory of planar differential systems”, Universitext, Springer-Verlag, Berlin, 2006, xvi+298pp (ISBN:3-540-32893-9).

Outline

  1. Examples of nonsmooth systems. Introduction to phase plane dynamics, simple ideas of flows and maps. Existence of equilibira, stability and bifurcations
  2. Piecewise linear maps in the plane, analysis and classification of equilibria, their existence, stability and bifurcations. Period adding bifurcations. Relationship to flows.
  3. Three dimensional piecewise linear maps; classification of simple bifurcations
  4. General n-dimensional piecewise linear maps.
  5. Other piecewise smooth maps in the plane (including ‘map with a gap’).
  6. Smoothing of piecewise linear maps and effect on period adding bifurcations.
  7. Application: global dynamics of low immersion high-speed milling.
  8. Nonsmooth flows; discontinuity induced bifurcations, Filippov system.
  9. One-parameter bifurcations of planar Filippov systems.
  10. Grazing and chattering; discontinuity mappings near grazing.
  11. Application: DC/DC converter.
  12. Towards a nonsmooth catastrophe theory (if time).

References

  • (For background in nonlinear dynamics) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Strogatz, S. H. Perseus Books.
  • One-parameter bifurcations of planar Filippov systems Kuznetsov, Y.A. et al Int. J. Bifurcation and Chaos 13, 2157- 2188 (2003).
  • Piecewise smooth dynamical systems: theory and applications. Di Bernardo et alSpringer, Applied Mathematical Sciences vol. 163.
 Hyperbolic Systems - Enrique Pujals (IMPA)

Outline

  1. Examples of hyperbolic sets: horseshoe, solenoid, Anosov, derivatives of Anosov and Plykin atractor.
  2. Transversal homoclinic intersections.
  3. Persistence and stability of hyperbolic sets; shadowing lemma.
  4. Stability of globally hyperbolic diffeomorphisms (Anosov).
  5. Filtrations and spectral decomposition of diffeomorphisms. Axiom A. Omega-stability theorem.
  6. Comments on the stability and omega-stability conjectures.
  7. Elements of bifurcation theory.
  8. Homoclinic tangencies. Newhouse phenomenon.
  9. Some notions on robust transitivity and partially hyperbolic systems.

References

  • A. Katok, B. Hasselblatt, Introduction to the moderm theory of dynamical systems; Cambridge University Press, 1995.
  • Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations; Cambridge University Press, 1995.
  • E. R. Pujals, M. Sambarino, Topics on homoclinic bifurcation, dominated splitting,robust transitivity and related results; Handbook of dynamical systems vol 1B, Elsevier (2005) 327-378.
  • M. Shub, Global Stability of Dynamical Systems; Springer-Verlag, 1987.
 V. Araujo, M. Viana
 Joan Torregrosa Arús, Universitat Autònoma de Barcelona (Spain)
 Maria Jesus Alvarez, Universitat de les Illes Balears (Spain)
 Antonio Esteban Teruel Aguilar, Universitat de les Illes Balears (Spain)
 Jorge Galan-Vioque, Universidad de Sevilla (Spain)
 Jordi Taixés, Universitat de Barcelona (Spain)
 Maite Grau, Universitat de Lleida (Spain)
 Ekaterina Kutafina, AGH University of Science and Technology (Poland)
 Set Pérez González, Universitat Autònoma de Barcelona (Spain)
 Pablo Gutiérrez Barrientos, Universidad de Oviedo (Spain)
 José Luis Bravo Trinidad, Universidad de Extremadura (Spain)
 Marta  Peña, Universitat Politècnica de Catalunya (Spain)
 Josep Ferrer, Universitat Politècnica de Catalunya (Spain)
 Rafael Obaya, Universidad de Valladolid (Spain)
 Ana María Sanz, Universidad de Valladolid  (Spain)
 Sergio Alejandro Carrillo Torres, Universidad Sergio Arboleda (Colombia)
 David  Blázquez Sanz, Universidad Sergio Arboleda (Colombia)
 Freek Verstringe, Hasselt University (Belgium)
 Magdalena Caubergh, Universitat Autònoma de Barcelona (Spain)
 Marina Gonchenko, Universitat Politècnica de Catalunya (Spain)
 Daniel Barajas Pichardo, Universitat Politècnica de Catalunya (Spain)
 Abraham De la Rosa Ibarra, Universitat Politècnica de Catalunya (Spain)
 Dmitry Vorotnikov, Voronezh State University  (Russia)
 Reginaldo Braz, Universidade Federal de Minas Gerais (Brazil)
 Naeem Alkoumi, Universidad de Granada (Spain)
 Leopoldo Morales López, Universitat Autònoma de Barcelona (Spain)
 Ashraf Owis, Cairo University (Egypt)
 Zhaoyang Dong, Universitat Autònoma de Barcelona (Spain)
 Angel Jorba, Universitat de Barcelona (Spain)
 Javier Ros Padilla, Universidad de Sevilla (Spain)
 Luciano  Coutinho dos Santos, Universidade Federal de Minas Gerais (Brazil)
 Fernando Fernández Sánchez, Universidad de Sevilla (Spain)
 Jordi Soria Comas, Universitat Politecnica de Catalunya (Spain)
 Adriana Buica, Universitatea Babes-Bolyai (Romania)
 Timofey Dragunov, University of Nizhny Novgorod (Russia)
 Víctor Muñoz-Villarragut, Universidad de Valladolid (Spain)
 Marcos Rodriguez, Universidad de Zaragoza (Spain)
 Patricia Yanguas Sayas, Universidad Pública de Navarra (Spain)
 Jesús Francisco Palacián Subiela, Universidad Pública de Navarra (Spain)
 Geraldo Cesar , Universidade Federal de Minas Gerais (Brazil)
 Clementa Alonso González, Universidad de Alicante (Spain)
 Eva Marina Pérez-Nueno, Universitat de Barcelona (Spain)
 Rubén Berenguel, Universitat de Barcelona (Spain)
 Mónica Molina Becerra, Universidad de Sevilla (Spain)
 Amadeu Delshams, Universitat Politècnica de Catalunya (Spain)
 Alejandro Luque, Universitat Politècnica de Catalunya (Spain)
 Marcel Guardia, Universitat Politecnica de Catalunya (Spain)
 Elisabeth García Medina, Universidad de Sevilla (Spain)
 Oswaldo Jose Larreal Barreto, Universitat Politecnica de Catalunya (Spain)
 Joan Carles Tatjer Montaña, Universitat de Barcelona (Spain)
 Jorge Álvarez López, Universidad de Valladolid (Spain)
 Carmen Núñez, Universidad de Valladolid (Spain)
 Alberto Llorente, Universidad de Valladolid (Spain)
 Lorena López Hernanz, Universidad de Valladolid (Spain)
 Jorge Mozo Fernández, Universidad de Valladolid (Spain)
 Pablo S. Casas, Universitat Politècnica de Catalunya (Spain)
 Jose Manuel Benita Bordes, Universitat Politècnica de Catalunya (Spain)
 José Angel Rodriguez Méndez, Universidad de Oviedo (Spain)
 Santiago Ibáñez, Universidad de Oviedo (SPAIN)
 Gemma Huguet Casades, Centre de Recerca Matemàtica (Spain)
 Pau Rabassa, Universitat de Barcelona (Spain)
 Albert Granados Corsellas, Universitat Politècnica de Catalunya (Spain)
 Soledad Fernández García, Universidad de Sevilla (Spain)
 Josep M. Olm, Universitat Politècnica de Catalunya (Spain)
 Mercelena  Hernandez Sierra, Corporación Universitaria de Colombia Ideas (Colombia)
 Ernest  Fontich, Universitat de Barcelona (Spain)
 Francesca di Girolamo, Università degli Studi di Milano Bicocca  (Italy)
 Oleg Makarenkov, Voronezh State University (Russia)
 Montse Navarro, Universitat de Barcelona (Spain)
 Sara Costa Romero, Universitat Autònoma de Barcelona (Spain)
 Enrique Ponce, Universidad de Sevilla (Spain)
 Francisco Torres, Universidad de Sevilla (Spain)
 Victoriano Carmona, Universidad de Sevilla (Spain)
 Begoña Alarcon, Universitat de València (Spain )
 Emilio Freire, Universidad de Sevilla (Spain)
 Andre Vanderbauwhede, Ghent University (Belgium)
 Gianluigi del Magno, Universidade de Lisboa (Portugal)
 Antonio Algaba Durán, Universidad de Huelva (Spain)
 Manuel Reyes Columé, Universidad de Huelva (Spain)
 Cristóbal García García, Universidad de Huelva (Spain)
 Daniel J. Pagano, Universidade Federal de Santa Catarina (Brazil)