Madrid, 23 January 2023 - 27 January 2023



This is the 19th RTNS (Recent Trends in Nonlinear Science) winter school in Dynamical Systems of the DANCE (Dinámica, Atractores y Nolinealidad: Caos y Estabilidad) Spanish network. We will start a new format with only two courses of 10 hours each. During afternoons, some of the participants will have the opportunity to present their recent works as short talks.

The RTNS2023 will take place at ICMAT in Madrid (Cantoblanco Campus).

Participants who want to present a short talk are requested to send, by email, a PDF file with the title and abstract using the CONTACT address.


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Joan Torregrosa (Universitat Autònoma de Barcelona)
Patricia Yanguas (Universidad Pública de Navarra)


María Jesús Álvarez Torres (Universitat de les Illes Balears)
Rosa María Benito Zafrilla (Universidad Politécnica de Madrid)
Gérard Iooss (Université de Nice)


Francisco Javier Arranz Saiz (Universidad Politécnica de Madrid)
Florentino Borondo Rodríguez  (Universidad Autónoma de Madrid - ICMAT)
Laia Domingo Colomer (Universidad Autónoma de Madrid - ICMAT)
Juan Carlos Losada González (Universidad Politécnica de Madrid)
Mar Grande Toledano (Universidad Politécnica de Madrid)
Fabio Revuelta Peña (Universidad Politécnica de Madrid)
Alberto Vergel Otero (Universidad Politécnica de Madrid - ICMAT)



Session I:

  1. Numerical computation of regularities of invariant objects with Daubechies wavelets
    Salvador Borrós Cullell (Universitat Autònoma de Barcelona, Spain)
  2. Local niteness of limit cycles for perturbations of linear non-degenerated centers in R3
    María Martín Vega (Universidad de Valladolid, Spain)
  3. Nilpotent centers on center manifolds in R3
    Lucas Queiroz Arakaki (São Paulo State University, Brazil)
  4. Relationship of the Green's functions related to the Hill's equation coupled to different boundary value conditions
    Mouhcine Yousfi (Universidade de Santiago de Compostela, Spain)

Session II:

  1. On a population model with density dependence and Allee effect
    Jose S. Cánovas (Universidad Politécnica de Cartagena, Spain)
  2. Flow map parameterization method for whiskered tori in quasi-periodic Hamiltonian systems
    Álvaro Fernández Mora (Universitat de Barcelona, Spain)
  3. Interaction between Hopf bifurcations in neuron models
    Diego Noriega Rodríguez (Universidad de Oviedo, Spain)

Session III:

  1. Poincaré-Hopf theorem for Filippov vector fields and Filippov singularity indices
    Joyce Aparecida Casimiro (State University of Campinas, Brazil)
  2. Critical transitions on nonautonomous d-concave population models
    Jesús Dueñas Pamplona (Universidad de Valladolid, Spain)
  3. Chaos and integrability from geometrodynamics
    Alberto Vergel Otero (Universidad Politécnica de Madrid-ICMAT, Spain)

Session IV:

  1. Higher order analysis in non-Lipschitz differential equations for periodic solutions using topological degree
    Francisco Bruno Gomes da Silva (State University of Campinas, Brazil)
  2. Limit cycles of some new families of phi-Laplacian Liénard equations
    Set Pérez Gonzalez (Universidad de Oviedo, Spain)
  3. Applications of the invariant manifolds in Chemistry
    Fabio Revuelta (Universidad Politécnica de Madrid, Spain)

Scientific Sessions

We will introduce several tools to study the number of limit cycles for families of ordinary differential equations (ODE), paying special attention on planar polynomial ODE and on one-dimensional periodic non-autonomous ones. These are the main subjects of the course:

  1. Poincaré map and local study of periodic orbits. Bifurcations of codimension one of limit cycles. Algebraic limit cycles.
  2. A miscellanea of useful tools: Chebyshev systems, rational parameterizations, Poincaré-Miranda theorem, Resultants.
  3. Upper bounds for the number of limit cycles: Lyapunov and Bendixson-Dulac approaches.
  4. Lower bounds for the number of limit cycles: Lyapunov constants, Abelian integrals and algebraic solutions.
  5. Abel type differential equations.
 A Journey in Nonlinear Dynamics Riding Optical Waves - Mariana Haragus (Institut FEMTO-ST, Université de Franche-Comté, France)

Starting from the mathematical questions raised by a problem in nonlinear optics, we present several tools used in the study of nonlinear waves. Focusing on bifurcations and stability, we discuss:

  1. The Lugiato-Lefever equation and its mathematical questions.
  2. Some elementary local bifurcations.
  3. Center manifolds and normal forms.
  4. Spatial dynamics and bifurcations of steady waves.
  5. Nonlinear dynamics of periodic waves.
 Jose S. Cánovas, Universidad Politécnica de Cartagena (Spain)
 Lluc Manuel Ferré Escobet, Universitat Autònoma de Barcelona (Spain)
 María Martín Vega, Universidad de Valladolid (Spain)
 Joyce Aparecida Casimiro, Universidade Estadual de Campinas (Brazil)
 Francisco Bruno Gomes da Silva, Universidade Estadual de Campinas (Brazil)
 Lucas Queiroz Arakaki, Universidade Estadual Paulista (Brazil)
 Miquel Barcelona Poza, Universitat Autònoma de Barcelona (Spain)
 Ainoa Murillo López, Universitat de Barcelona (Spain)
 Carles Trullàs Fernàndez, Universitat Politècnica de Catalunya (Spain)
 Salvador Borrós Cullell, Universitat Autònoma de Barcelona (Spain)
 Diego Noriega Rodríguez, Universidad de Oviedo (Spain)
 Ernesto Lanchares Sánchez, Universidad de La Rioja (Spain)
 Mónica Molina Becerra, Universidad de Sevilla (Spain)
 Juan R. Sisternes Alacot, Universidad Politécnica de Madrid (Spain)
 Enrique Ponce Núñez, Universidad de Sevilla (Spain)
 Jesús Dueñas Pamplona, Universidad de Valladolid (Spain)
 Víctor  Lanchares Barrasa, Universidad de La Rioja  (Spain)
 Fernando Fernández Sánchez, Universidad de Sevilla (Spain)
 Fernando Sanz Sánchez, Universidad de Valladolid (Spain)
 Clementa Alonso González, Universidad de Alicante (Spain)
 Isaac  García, Universitat de Lleida (Spain)
 Susanna Maza, Universitat de Lleida (Spain)
 Marina Esteban, Universidad de Córdoba (Spain)
 Juan José Morales Ruiz, Universidad Politécnica de Madrid (Spain)
 Álvaro  Fernández Mora, Universitat de Barcelona (Spain)
 Mouhcine Yousfi, Universidade de Santiago de Compostela (Spain)
 Set Pérez González, Universidad de Oviedo (Spain)
 Javier Ros Padilla, Universidad de Sevilla (Spain)
 Elísabet Vela Felardo, Universidad de Córdoba  (Spain)
 Joan Torregrosa, Universitat Autònoma de Barcelona (Spain)
 Armengol Gasull, Universitat Autònoma de Barcelona (Spain)
 Mariana Haragus, Université de Franche-Comté (France)
 Francisco Javier Arranz, Universidad Politécnica de Madrid (Spain)
 Florentino Borondo, Universidad Autónoma de Madrid (Spain)
 Arturo Vieiro Yanes, Universitat de Barcelona (Spain)
 Rosa M. Benito, Universidad Politécnica de Madrid (Spain)
 Fabio Revuelta, Universidad Politécnica de Madrid (Spain)
 Juan Carlos Losada, Universidad Politécnica de Madrid (Spain)
 Alberto Vergel, Universidad Politécnica de Madrid (Spain)
 Luis Ángel Calderón Pérez, Universitat de les Illes Balears (Spain)