• Joan Gimeno (Universitat de Barcelona)
  New Solutions from Functional Perturbed Uniformly HyperbolicTrajectories
  Resumen: We develop a method to construct solutions of some (retarded or advanced) equations. We assume that the equations considered are formally close to an ODE and that the ODE admits hyperbolic solutions (that is, perturbations transversal to a trajectory grow exponentially either in the past or in the future) and we show that there are solutions of the functional equation close to these hyperbolic solutions of the ODE. The method of proof does not require to formulate the delayed problem as an evolution for a class of initial data.  The main result is formulated in an "a-posteriori" format and allows to show that solutions obtained by non-rigorous approximations are close (in some precise sense) to true solutions. A prime application is on the motion of point charges interacting via the fully relativistic Lienard-Wiechert potentials (as suggested by J.A. Wheeler and R.P. Feynman in the 1940's).  These are retarded equations, but the delay depends implicitly on the trajectory. In this electrodynamics (or gravitational) case, our result allows to compare the hyperbolic solutions of several post-newtonian approximations or numerical approximations with the solutions of the Lienard-Wiechert interaction. A similar construction is deduced to compute approximate solutions by using Formal Power Series (FPS) on the geometric framework of Normally Hyperbolic Invariant Manifolds (NHIMs) where by systematically deforming the original solutions and modulating slow variables, we construct perturbative expansions that remain valid under mild regularity conditions. This is a joint finished and ongoing works with R. de la Llave and J. Yang.
  
  
• Begoña Nicolás (Universidade de Santiago de Compostela)
  High order parametrization of invariant manifolds to design capture of asteroids
  Resumen: Hyperbolic invariant manifolds are sometimes proposed to be used in the capture of asteroids, specially those that are near Earth, relying on the dynamics of invariant objects to keep the asteroid confined in a specific region. In this talk, we will talk about how the hyperbolic invariant manifolds associated with the horizontal family of 2D invariant tori around L3 in a Bicircular Sun-perturbed Earth-Moon system reach different locations inside and outside the Earth-Moon vicinity. First, we will use the fundamental domains of the stable manifolds to identify the sets of trajectories that surround the position of the asteroid at different times and that can be used to lead the asteroid towards L3. Then, we will be able to compute the exact initial condition on the fundamental domain of the stable invariant manifold that lies as close as we want of the position of the asteroid at a given time, as well as the instantaneous cost in ∆v that is necessary to inject the asteroid into the stable manifold. In order to perform this computation, an accurate approximation of the invariant manifolds is required. For this, we have implemented a high order approximation of the invariant manifolds of invariant tori, based on the parametrization method combined with the jet transport technique.
  
  

• Carmen Mayora-Cebollero (Universidad de Zaragoza)
  Deep learning for dynamical systems analyses

  Resumen: Dynamical systems are usually analysed using standard techniques such as Lyapunov exponents. However, most of these classical methods are computationally expensive and often not feasible for studying real-world data. In this communication, we propose to use Deep Learning to overcome such limitations. We apply Deep Learning to detect chaotic regions in the parameter space of classical dynamical systems and to analyse chaotic dynamics in biological time series. Furthermore, we use Deep Learning to detect more complex dynamical regimes by approximating Lyapunov exponents from single-variable time series. These analyses show the potential of Deep Learning in the study of dynamical systems. Joint work with Roberto Barrio, Álvaro Lozano, Ana Mayora-Cebollero, Sergio Serrano, and Rubén Vigara (Universidad de Zaragoza, Spain).

   

•  Josep-María Mondelo (Universitat Autònoma de Barcelona)
  The parameterization method in the exploration of the dynamics of the RTBP.
  Resumen: The circular, spatial Restricted Three-Body Problem is the basic model used in preliminary mission design of libration point missions. The nominal trajectories of these missions are contained in the center manifold of the collinear libration points. The hyperbolic behaviour around these points can be used to obtain transfer trajectories to/from a body or connecting different nominal trajectories. In this talk, we will review some of the dynamics around the collinear libration points and some of the methodologies to computationally describe it in a global manner. The emphasis will be on advances from the last decade in the computational use of the parameterization method, both in a semi-analytical and a purely numerical manner.
  
  

RESUMEN : En esta sesión se presentan resultados recientes cuyos nexos comunes son los sistemas dinámicos y su aplicabilidad a la modelización de fenómenos o procesos reales. Por otra parte, las técnicas empleadas por cada autor, así como el tipo de resultados obtenidos, son muy diferentes, considerándose problemas discretos, continuos e incluso problemas con impulsos.

• Roberto Barrio (Universidad de Zaragoza)
  Análisis fast-slow y bifurcaciones en dinámica cardíaca
  RESUMEN: Las depolarizaciones posteriores tempranas (Early Afterdepolarizations, EADs) son comportamientos anormales que pueden conducir a la insuficiencia cardíaca e incluso a la muerte cardíaca. En esta presentación, investigamos matemáticamente la aparición y el desarrollo de estos fenómenos en dos modelos realistas de miocitos ventriculares: el modelo de Sato (2009), conejo, y el modelo humano de O'Hara (2011). Conectamos los resultados obtenidos en dichos modelos de alta dimensión con un modelo cardíaco de Luo-Rudy  de dimensión baja. Examinando la estructura de bifurcaciones del modelo de Luo-Rudy, se describen los elementos dinámicos asociados con los patrones y transiciones observados en los modelos realistas. Utilizando un análisis "fast-slow", exploramos la aparición y evolución de EADs en el modelo de baja dimensión y desarrollamos nuevas metodologías para la descomposición "fast-slow" para el modelo realista de O'Hara en dimensión superior.
• Daniel Franco (Universidad Nacional de Educación a Distancia)
  Modelos de población con dispersión
  RESUMEN: El hábitat de muchas especies está fragmentado, por lo que es crucial comprender cómo la dispersión entre las distintas regiones que lo componen impacta en, por ejemplo, el tamaño total de la población. En esta presentación discutiremos resultados recientes que estudian, en distintas situaciones, cómo responde el tamaño total de la población ante un aumento de la tasa de dispersión.
• F. Adrián F. Tojo (Universidade de Santiago de Compostela)
  La unificación de la dinámica continua y discreta: una introducción al cálculo de Stieltjes
  RESUMEN:Diez años después de la publicación de "A New Unification of Continuous, Discrete, and Impulsive Calculus through Stieltjes Derivatives", el cálculo de Stieltjes ha experimentado notables avances, a la vez que ha planteado nuevos e importantes desafíos. En esta charla exploraremos el origen y los fundamentos del cálculo de Stieltjes, su conexión con otras teorías que integran dinámicas continuas y discretas, y destacaremos sus aplicaciones más relevantes, así como las perspectivas y retos que marcan su desarrollo futuro.
• Antonio J. Ureña (Universidad de Granada)
  Perturbaciones de un continuo de soluciones periódicas
  RESUMEN: Consideremos un sistema Hamiltoniano dependiente periódicamente del tiempo. Si hay una solución periódica no degenerada, el teorema de la función implícita permite encontrar soluciones periódicas cercanas para pequeñas perturbaciones del sistema. Pero este método no se aplica a problemas con un continuo de soluciones periódicas, ya que estas han de ser forzosamente degeneradas. No obstante, en ciertos casos hay técnicas que permiten probar la persistencia de algunas soluciones periódicas. En esta charla exploramos la validez de este tipo de resultados dependiendo de cómo se entiendan las perturbaciones. Este es un trabajo conjunto con Rafael Ortega.

RESUMEN: "Dinámica de Atractores No Caóticos Extraños". El interés en el estudio de los ANCEs fue la base de la creación de nuestra red, hace más de 23 años. Estos objetos invariantes, intrínsecos de  la dinámica no autónoma, presentan una dinámica con nivel muy alto de complejidad  y están en general muy lejos de ser persistentes por pequeñas variaciones en el correspondiente sistema dinámico. Los resultados teóricos que respaldan las evidencias numéricas sobre su existencia para distintas ecuaciones diferenciales o mapas son escasos y  complicados de obtener, por lo que los avances en esta rama de las matemáticas son lentos. Esta sesión recuerda el interés de los ANCEs, con resultados recientes sobre sistemas dinámicos en los que la aparición de tales objetos es posible.

Conferenciantes invitados:

•  Lluis Alsedà (Universitat Autònoma de Barcelona)
  

  On the contribution of semianalytic approximations of invariant curves to the dynamical study of quasiperidic forced maps in the annulus
  Resumen: We will explore how semianalytic approximations to invariant curves of quasiperidic forced maps in the annulus may help in studying the dynamics of the system. More concretely, in checking whether an invariant curve is a strange non-chaotic attractor and also, time permitting, we will study bifurcations in quasiperidic forced maps in the annulus.

• Amadeu Delshams (Universitat Politècnica de Catalunya)
  

  A reduction method for invariant curves of quasi-periodically forced maps
  Resumen: The existence of translated curves for quasi-periodically forced maps is established, under very mild regularity hypotheses, for rotation numbers of constant type. Among translated curves, invariant curves are characterized as solutions of a scalar bifurcation equation, from which their existence, stability and bifurcation can be easily described. This talk is based on an unfinished joint work with Rafael Ortega, U. Granada.

• Rafael Obaya (Universidad de Valladolid)
  

  Atractores extraños y dinámica caótica en EDOs escalares no autónomas. La transición de una a varias medidas ergódicas
  Resumen: Consideramos ecuaciones diferenciales escalares con variación temporal casiperiódica que poseen atractores extraños. Analizaremos propiedades de complejidad del flujo asociadas al carácter sensitivo respecto de condiciones iniciales, a la existencia de varias medidas ergódicas y de exponentes de Lyapunov positivos. Mostraremos ciertos tipos de ecuaciones casiperiódicas de tipo lineal-disipativo que admiten un atractor caótico en el sentido de Devaney. Posteriormente, estudiaremos las versiones de las nociones anteriores en ecuaciones diferenciales escalares cuya variación temporal admite varias medidas ergódicas. Aparecen escenarios dinámicos nuevos que no son posibles en los modelos casiperiódicos. En particular, explicaremos patrones de bifurcación pitchfork generalizada, específicos de este contexto “multiergódico”.

• Joan Carles Tatjer (Universitat de Barcelona)
  

  Bifurcaciones no suaves en aplicaciones lineales a trozos forzadas casiperiódicamente y ANCEs
  Resumen: El objetivo de esta charla es ilustrar como aparecen atractores no caóticos extraños (ANCE) en familias de aplicaciones forzadas casiperiódicamente en modelos lineales a trozos de bifurcaciones silla-nodo y pitchfork
  no suaves, vía fractalización de curvas invariantes. En estos casos, se puede demostrar de manera rigurosa la existencia de tales atractores para determinados valores de los parámetros.

Conferenciantes invitados:

• Jorge Galán (Universidad de Sevilla)
  Degenerate Subharmonic Bifurcations of Quasi Satellite Orbits in the Mars-Phobos Spatial Circular Restricted Three Body Problem
  

  We present a  methodology to continue  and analyze the bifurcation behaviour of periodic orbits for Quasi Satellite Orbits (QSO) in the Spatial Circular Restricted Three Body  Problem for the Mars Phobos
  system. In particular, we concentrate on degenerate subharmonic bifurcations where the elliptic Floquet multipliers stop on the unit circle and turn around passing twice for a given q-subharmonic bifurcation. It is  based on a continuation method for periodic orbits in the presence of conserved quantities. These orbits have received renewed interest for survey missions on the Moon,
  Martian moons, and asteroids in general. The complete exploration of all QSO is an unending task,
  and here we focus on three particular families characterized by the number of revolutions around the satellite of the base QSO. We have detected several bifurcations to nonplanar close to stable solutions
  that might be relevant for Phobos observation missions.

  Work in collaboration with J.M Montilla, C. Bombardelli and A. Martinez-Cacho.

  
  
  
• Miguel Garrido (Universitat Autònoma de Barcelona)
  Parabolic saddles and Newhouse domains in Celestial Mechanics
  McGehee introduced a compactification of the phase space of the restricted
  3-body problem by gluing a manifold of periodic orbits “at infinity”. Although
  from the dynamical point of view these periodic orbits are parabolic (the lin-
  earization of the Poincaré map is the identity matrix), one of them, denoted
  here by O, possesses stable and unstable manifolds which, moreover, separate
  the regions of bounded and unbounded motion.
  This observation prompted the investigation of the homoclinic picture asso-
  ciated to O, starting with the work of Alekseev and Moser. We continue this
  research and extend, to this degenerate setting, some classical results in the
  theory of homoclinic bifurcations. More concretely, we prove that there exist
  Newhouse domains N in parameter space (the ratio of masses of the bodies)
  and residual subsets R ⊂ N for which the homoclinic class of O has maximal
  Hausdorff dimension and is accumulated by generic elliptic periodic orbits.
  One of the main consequences of our work is the fact that, for a (locally)
  topologically large set of parameters of the restricted 3-body problem the union
  of its elliptic islands forms an unbounded subset of the phase space and, more-
  over, the closure of the set of generic elliptic periodic orbits contains hyperbolic
  sets with Hausdorff dimension arbitrarily close to maximal. Other instances of
  the restricted n-body problem such as the Sitnikov problem and the case n = 4
  are also considered.
  This is a joint work with Pau Martı́n and Jaime Paradela.
  
  
• Víctor Lanchares (Universidad de La Rioja)
  On the restricted problem of 2 + n bodies
  Resumen: The restricted problem of 2 + n bodies was introduced by A. L. Whipple and V. Szebehely [1] as a generalization of the well known circular restricted three body problem. The motion of n infinitesimal masses is considered under the attraction of two principal bodies which move in a circular orbit. In this talk we consider some frameworks where this problem can be of special interest as well as some questions regarding equilibrium solutions and their stability.
  [1] A. L. Whipple, V. Szebehely. The restricted problem of n + ν bodies. Celestial Mechanics, 32, 137–144, 1984. 
  
  
  
• Daniel Casanova (Universidad de Zaragoza)

  Dynamical modeling and characterization of Space Debris 
  Resumen: Space debris represents a growing threat to operational satellites and space missions, especially in heavily populated orbital regions such as Low Earth Orbit (LEO), Medium Earth Orbit (MEO), and Geostationary Orbit (GEO), including the dynamically complex Laplace Plane. In this talk, it is presented a comprehensive overview of the current space debris environment across these orbital zones and the dynamical models that govern object evolution. Observational data are used to derive orbital elements or state vectors, which are then numerically propagated to assess the temporal evolution of individual objects. Clustering algorithms are employed to identify groups of objects with similar orbital

  behavior, designating a 'leader' and its associated members. Once such families are identified, we apply an analytical orbital propagator to efficiently study their evolution. This approach allows for a parametric analysis of the area-to-mass ratio, facilitating its estimation by fitting the analytical model to the observed trajectories.

• Ana Mayora-Cebollero (Universidad de Zaragoza)
  Título de la tesis: Coupled Dynamical Systems: Applications in Neural Networks and Biomathematics
  Director: Roberto Barrio
  Resumen: The study of coupled systems is of great interest as it allows us to understand and analyze many phenomena in nature. These phenomena can be described mathematically as the interaction between elementary dynamical units. The classical examples of coupled systems are related to chemical reactors and neural networks. With respect to the former example, we study the dynamics of two coupled Brusselators. Although the dynamics of an isolated Brusselator is very simple, complex dynamics as chaos emerges when coupling two Brusselators. In the literature, there exist two chaotic zones located in different regions of the parameter space. The question that arises is whether both zones are connected or not. We define a new linking parameter that connects both biparametric regions containing these zones and we continue the period-doubling bifurcations generating one of the chaotic regions to conclude that they are not connected. Thanks to this study, a new chaotic zone has been located. In this system of two coupled Brusselators, there are regions of the parameter space in which the Brusselators are almost synchronized during most of the time, except for brief intervals when their synchronization is lost. This phenomenon is analyzed in the system of two and three coupled Brusselators. Regarding the latter example of coupled systems, we first study the dynamics of a Central Pattern Generator (CPG) model. We observe that the bursting is maintained in this model even when this type of behavior has disappeared for the model of an isolated neuron. The persistence of the bursting region leads to an increase of the area characterized by the predominance of the tripod gait pattern. Moreover, some hyperchaotic zones have been detected. Later, we analyze the dynamics of two mean-field models of coupled neural populations. One of these models (the model of Dumont and Gutkin) considers synaptic dynamics, while the other (the model of Montbrió, Pazó and Roxin) does not. Both are connected through a parameter related to the synapsis and the model of Montbrió et al. is a limit case of the model of Dumont and Gutkin. There exist chaotic zones when there is no synaptic dynamics or it is very weak. However, when there is synaptic dynamics, chaos is not present in the studied region and bursting-type dynamics may appear. The disappearance of chaos is explained through the numerical study of bifurcations and the concept of geometric bifurcations. Finally, we study the dynamics of a mean-field model with a sufficiently rich structure of bifurcations capable of emulating characteristic dynamical behaviors of diseases as epilepsy.
  
  
• Álvaro Fernández-Mora (Universitat de Barcelona)
  

  Título de la tesis: Flow map parameterization methods for invariant tori in Hamiltonian systems
  Director: Alex Haro and Josep-Maria Mondelo
  Resumen: The goal of this thesis is to advance the development of Kolmogorov–Arnold–Moser (KAM)-type techniques within the framework of the parameterization method and their application to problems in celestial mechanics. We have developed efficient iterative KAM schemes for computing  partially hyperbolic invariant tori and their invariant manifolds in quasi-periodic Hamiltonian systems. The efficiency arises from the geometric properties of the phase space (i.e., symplectic geometry), the systems (exact symplecticity), the tori (isotropy, Lagrangianity), and dynamical properties (reducibility). We also obtained an a posteriori theorem for the tori we   consider and their rank-1 invariant bundles. The approach followed allows applying the theorem to autonomous, periodic, and quasi-periodic Hamiltonian systems, and constitutes a proof of convergence for methods based on flow maps.  Furthermore, the theorem is suitable to conduct computer-assisted proofs of existence of invariant tori. The algorithms have been implemented and applied to the Circular and Elliptic Restricted Three-Body Problem (ERTBP) to compute an extensive set of non-resonant invariant tori along with their invariant bundles and whiskers.

• Jesus Dueñas (Universidad de Valladolid)
  Título de la tesis: D-concave nonautonomous differential equations and applications to critical transitions
  Director: Carmen Núñez y Rafael Obaya
  Universidad: Universidad de Valladolid
  Resumen: The results presented in this document delve deeper into nonautonomous bifurcation theory with a view towards critical transitions. Nonautonomous d-concave scalar differential equations have been studied due to their significance in modeling various real-world phenomena. Special interest has been placed on their applications in ecology, where d-concave equations are frequently employed to describe single species populations subject to the Allee effect.

• Luis Ángel Calderón Pérez (Universidad de Extremadura)
  Título: Curvas invariantes y ciclos límite en ecuaciones de Abel
  Director: José Luis Bravo e Ignacio Ojeda Martínez

  Resumen: el problema de Smale-Pugh consiste en dar cortas para el número de ciclos límite de una ecuación de Abel x'=A(t)x^3+B(t)x^2+C(t) con coeficientes periódicos. La fuerte relación de estas ecuaciones con ciertas familias de sistemas planos hace que en estos casos el problema de Smale-Pugh sea parte del problema XVI de Hilbert. J. Huang, H. Liang y J. Llibre obtienen una generalización de resultados clásicos de este problema utilizando la presencia de una curva invariante de cierto tipo en la ecuación de Abel. El objetivo de la tesis consiste en mejorar estos resultados, y obtener información general sobre el número de curvas invariantes de este tipo que una ecuación de Abel podría tener, con el objetivo de generalizar aún más estos resultados si la ecuación tiene más de una curva invariante.

• Miquel Barcelona (Universitat Autònoma de Barcelona)
  Título: Systematic Approaches to the Computation of Heteroclinic Connections between the Center Manifolds of the Collinear Libration Points in the Restricted Three-Body Problem
  Director: Josep-Maria Mondelo y Alex Haro
  

  Resumen: Heteroclinic connections in the spatial, circular, Restricted Three-Body Problem play are well recognized for their significant role in enabling zero-propellant transfer opportunities. These connections, which occur at the intersections between hyperbolic manifolds of invariant sets, are crucial not only for mission design but also for understanding the system's global dynamical behavior. In this work we present a systematic methodology for computing heteroclinic connections between saddle-center fixed points of 3dof, autonomous Hamiltonian systems. We also provide results on the geometric structure of the heteroclinic manifolds and discuss potential applications in mission design.

• Juan Pello (Universitat de Barcelona)
  Título de la tesis: Degenerate invariant tori in KAM theory
  Director: Alex Haro Provinciale/ Ernest Fontich Julià
  Universidad: Universitat de Barcelona
  Resumen: The thesis develops an incipient methodology to study bifurcations of invariant curves in quasi–periodic forced skew–product dynamical systems. In these discrete systems the phase space is a bundle in which the base is the torus T = R/Z, and the fiber is the real line. This approach involves KAM theory, translated graph theorems and bifurcations theory in the spirit of Moser, Herman, Rüßmann, Delshams and Ortega. In the project, rigorous results on the existence of parametric families of invariant translated tori are obtained in an a posteriori format and the methodology to study their bifurcations established. This a posteriori format is suitable to develop numeric calculations. Thus, the algorithms derived from the KAM iterative procedure associated to this methodology have been performed and implemented on the computer. The first part of the work is devoted to obtain sufficient conditions to find invariant tori, proving the existence and the analyticity of these invariant curves by means of an iterative procedure whose methodology involves KAM theory. In addition to the assumed non-resonance condition related to the Diophantine character of the frequency, certain nondegeneracy conditions are necessary to guarantee the constructability of the procedure and their convergence. In the context of analytic functions, formal solutions to the invariance equation, expressed by means of their Fourier series expansions, can be find. The convergence of these Fourier series leads to the problem of small divisors and cohomological equations. In the spirit of Delshams and De la Llave , the so–called translated graph method is adopted in this framework. The method consists essentially of fixing an average p ∈ R (in addition to the Diophantine frequency ω, which is also fixed a priori) and finding p–average invariant translated curves following a Newton–like iterative procedure. Before dealing with the translated graph method and the KAM procedure, the concept of reducibility of a skew-product is discussed. It is shown, as it is well known, that every non–singular one–dimensional linear quasi–periodic skew–product is reducible. This fact, and the explicit expression of the Floquet transformation that relates the reducibility with the Lyapunov exponents by means of the cohomological operator is used later to simplify some computations and obtain important dynamical properties. The first way to build the KAM iterative procedure elapses simultaneously combining two procedures. On the one hand, the invariance of the translated curve and, on the other hand, the reducibility of the skew–product without using the aforementioned explicit expression of the reducibility function (Floquet transformation).  Briefly speaking, the following result is proved: If we have a good enough approximation of a translated invariant curve, then under certain non–degeneracy and non–resonance conditions, there exists a true invariant translated curve nearby. The whole process is performed, starting with the non–degeneracy conditions needed for one step, the corresponding estimates, the iterative lemma, and finally, the KAM  theorem, in which the convergence and the analyticity of the solutions are proved. This is one of the most important results obtained in this work. Additionally, a second way to find invariant translated curves is proposed assuming the reducibility of the linearized system, avoiding the error produced in the procedure due to the reducibility part. These central results give rise to a methodology to study bifurcations of invariant curves analyzing the zeros of the translation function with respect to the parameters. In particular, saddle-node, transcritical, pitchfork and period–doubling bifurcations are discussed. The implementation of numerical procedures that allow validating the theoretical results, as well as reinforcing the numerical results obtained by applying them to specific examples had been covered in the last part of the thesis.

• Salvador Borrós (Universitat Autònoma de Barcelona)
  Título de la tesis: Obtención de coeficientes de ondículas de QPFSP
  Director: Lluís Alsedà
  Universidad: Universitat Autònoma de Barcelona
  Resumen: Discutiremos como obtener expansiones de ondículas truncadas de atractores en quasi-periodically forced skew products. En particular nos centraremos en los detalles de implementación y la calidad de la aproximación, especialmente cuando aproximamos atractores no caóticos estraños.