GSDUAB Working Seminar

The talks take place on Mondays at 15:00 (BCN time, CET/CEST) at the UAB seminar C1/-128.

Next talk

  • May 19, 2025, 15:00  
    Miquel Barcelona (Universitat Autònoma de Barcelona, Spain)
    Title: TBA.

    Abstract: TBA.

Upcoming talks

  • May 05, 2025, 15:00  No seminar scheduled
  • May 12, 2025, 15:00  No seminar scheduled
  • May 19, 2025, 15:00  Miquel Barcelona (Universitat Autònoma de Barcelona, Spain)
  • May 26, 2025, 15:00  Miguel Garrido (Universitat Autònoma de Barcelona, Spain)
  • June 2, 2025, 15:00  J. Tomás Lázaro (Universitat Politècnica de Catalunya, Spain)

Past talks

  • March 31, 2025, 15:00  
    Daniel Pérez Palau (Universitat Politècnica de Catalunya, Spain)
    Title: Predating in metapopulations: when $1+1\neq 2$ and other counterintuitivities.

    Abstract: Population models have claimed the attention of researchers on dynamical systems from the beginning of the discipline. During the last years, coupled logistic equations have been used to model the growth of a single species that lives in different (but connected) environments (or patches). In population dynamics it is also typical to study predator-prey models in which both species interact between each other. In this talk we will revisit classical results obtained in both models with and without predators. Then we will keep exploring these systems. We will encounter some situations that will defy our initial intuition in the topic.

  • March 24, 2025, 15:00  
    Joan Carles Artés (Universitat Autònoma de Barcelona, Spain)
    Title: Codimension in planar polynomial differential systems.

    Abstract: In this work we are concerned with the concept of  codimension for planar polynomial vector fields. This notion occurs in various articles applied to specific cases such as for example a cusp singularity of codimension 2 or a codimension 1 homoclinic loop of a saddle point with non-zero trace. We needed a rigorous, general definition that could be used efficiently not just for small codimensions as it usually occurs in the literature. For global problems such as the topological classification problem of planar quadratic vector fields modulo limit cycles we need to deal with higher codimensions and  handling higher codimensions becomes quite challenging as  the number of possibilities increases. We extend Sotomayor's definition  based on the topological  equivalence relation, allowing other equivalence relations such as for example the geometric one. We need to apply it  for more general objects than singularilies or homoclinic loops, as global configurations of singular points,  more general graphics and even phase portraits. Finally we apply our extended definition of codimension to assign a specific codimension to all 207 global topological configurations of singularities of quadratic differential systems excepting those with centers.

  • March 17, 2025, 15:00  
    Sebastián Buedo (Universidade de Santiago de Compostela, Spain)
    Title: The influence of discrete-time dynamics in the study of delay differential equations.

    Abstract: When it comes to modeling natural phenomena whose related magnitudes evolve according to certain past states, a system of scalar ordinary differential equations might not be an appropriate tool. Instead, delay differential equations (DDEs) constitute an adequate framework in order to tackle such a task. Qualitatively speaking, one key property of DDEs is that the nature of their underlying phase space is infinite-dimensional and thus highly complicated behaviour may appear. Nevertheless, there exists a family of DDEs such that the asymptotic properties of their solutions can be studied via one-dimensional difference equations. Indeed, this is the case of $x'(t)=-x(t)+f(x(t-\tau))$, given by certain function $f$ and $\tau>0$, an equation related to the dynamics of production and destruction (as stated by U. an der Heiden and M.C. Mackey), which can be studied through, e.g., $x_{n+1}=f(x_n).$

    In this talk, I will introduce DDEs, explain some of their basic properties and then discuss how the above-mentioned family can be studied through discrete-time dynamics. Finally, I will share with the audience which are those problems and questions that somehow stem from the former ideas and that partially fostered my research visit at UAB. In particular, I will explain some lines I have been working in with my colleague D. Cao Labora.
  • March 10, 2025, 15:00  
    Daniel Cao (Universidade de Santiago de Compostela, Spain)
    Title: Jack of some trades, master of none. Some open problems.

    Abstract: In this talk, I will present my brief scientific trajectory, entirely framed within the past ten years. To do so, I will highlight some of the contributions I have made during this time, as well as the mathematical techniques I have employed. In particular, I will mention various open mathematical problems that I have encountered and consider interesting. To facilitate the flow of the talk, the presentation of the more technical aspects will be kept self-contained and concise.
    Some of the topics I will address in the talk include: the fundamental theorem of algebra in quaternions, various issues related to fractional calculus, sufficiency criteria for a given function to define a distance, Borwein integrals, an elementary (failed) proof of Picard’s theorem, and another unfinished one...

  • March 03, 2025, 15:00  
    Gabriel Rondon (Universitat Autònoma de Barcelona, Spain)
    Title: On limit cycles in regularized Filippov systems bifurcating from homoclinic-like connections to regular-tangential singularities.

    Abstract: In this work, we are concerned with the smoothing of Filippov systems around homoclinic-like connections to regular-tangential singularities. We provide conditions to guarantee the existence of limit cycles bifurcating from such connections. Additional conditions are also provided to ensure the stability and uniqueness of these limit cycles. All the proofs are based on the construction of the first return map of the regularized Filippov system around homoclinic-like connections. This map is obtained by using a recent characterization of the local behavior of the regularized Filippov system around regular-tangential singularities. Fixed point theorems and Poincaré–Bendixson arguments are also employed.

  • February 24, 2025, 15:00 
    Douglas Duarte Novaes (Universidade Estadual de Campinas, Brazil)
    Title: Degenerate Parabolic-Parabolic Singularities in Filippov Vector Fields: Lyapunov Coefficients and the Generalized Pseudo-Hopf Bifurcation.

    Abstract: The cyclicity problem consists in estimating the number of limit cycles emanating from monodromic singularities.  Traditionally, this estimation relies on Lyapunov coefficients. However, in nonsmooth systems, besides the limit cycles bifurcating by varying the Lyapunov coefficients, monodromic singularities on the switching curve can always be split apart yielding, under suitable conditions, a sliding region and an additional limit cycle surrounding it. This bifurcation phenomenon, known as pseudo-Hopf bifurcation, has enhanced lower-bound cyclicity estimations for monodromic singularities in Filippov systems. Here, we push beyond the pseudo-Hopf bifurcation, demonstrating that the destruction of (2k) degenerate parabolic-parabolic singularities yields at least k limit cycles surrounding sliding segments.

  • February 17, 2025, 15:00
    Luis Fernando Mello (Universidade Federal de Itajubá, Brazil)
    Title:  The matching of two Markus-Yamabe crossing piecewise smooth systems in the plane.

    Abstract: A planar Markus-Yamabe vector field is a smooth vector field in the plane having only one equilibrium point and such that the spectrum of its Jacobian matrix at any point is on the left of the imaginary axis in the complex plane. A vector field is globally asymptotically stable if it has a globally asymptotically stable equilibrium point p: all the orbits tend to p in forward time. One of the great results of the Qualitative Theory of Differential Equations establishes that a planar Markus-Yamabe vector field is globally asymptotically stable. We prove that planar crossing piecewise smooth vector fields defined in two zones formed by two Markus-Yamabe vector fields sharing the same equilibrium point located on the separation straight line are not necessarily globally asymptotically stable. This is a study in collaboration with D.C. Braga, F.S. Dias and J. Llibre.

  • February 10, 2025, 15:00
    Jean-Marc Ginoux (Université de Toulon, France)
    Title:  Energy function of 2D & 3D dynamical systems (40min) & History of special relativity theory: 120 years on (20min)

    Abstract: [First talk] The aim of this talk is to prove that the jerk equation of three-dimensional autonomous dynamical systems must be multiplied by the second time derivative of the state variable and not by the first like in dimension two. We then provide an interpretation of the new term appearing in the energy function and called jerk energy. Two and three-dimensional Van der Pol models are then used to exemplify these main results. Applications to Lorenz and Chua's models confirms their validity. Then, we will focus on the energy function of two and three-dimensional coarse systems and prove that it can be directly deduced from the so-called flow curvature manifold. 
    [Second talk] The aim of this talk is to show that as early as may 1905 (before Einstein had published his famous article) Henri Poincaré has already laid the foundations of the special relativity theory. 

  • January 27, 2025, 15:00
    Otavio Henrique Perez (Universidade de São Paulo, Brazil)
    Title:  Limit cycles of canard type of non linear regularizations of piecewise smooth vector fields

    Abstract: Let $Z$ be a planar piecewise smooth vector field and denote its switching boundary by $\Sigma$. Suppose that $p\in\Sigma$ is a fold-fold singularity. We study limit cycles of nonlinear regularizations of $Z$ around $p$ in the case in which $\Sigma\backslash\{p\}$ consists of sewing points. We give a simple criterion for lower bounds and the existence of limit cycles of canard type, expressed in terms of zeros of slow divergence integral. Using the criterion, we prove that it there exist a quadratic regularization of non-smooth center such that, for any integer $k>0$, it has at least $k$ hyperbolic limit cycles, for a suitably chosen monotonic transition function $\varphi_k:\mathbb{R}\rightarrow\mathbb{R}$ and $\varepsilon > 0$ sufficiently small. We prove a similar result concerning codimension 1 non-smooth focus II$_2$, providing more limit cycles than the known results concerning the Sotomayor--Teixeira regularization. This is a work in progress with Peter de Maesschalck (UHasselt) and Renato Huzak (UHasselt).

  • September 2, 2024, 15:00
    Melvin Yeung (Hasselt University, Belgium)
    Title:  A maximum modulus theorem for functions admitting Stokes phenomena applied to the Problem of Dulac

    Abstract: In this talk we will discuss a finiteness result for limit cycles close to a certain class of polycycles. For this we will use techniques of Ilyashenko and the class of polycycles is chosen such that there are no problems with asymptotics. During the talk we will introduce a very general version of Stokes phenomena and discuss a version of Maximum Modulus Theorem for functions admitting Stokes phenomena.

  • June 17, 2024, 15:00
    Paulo Henrique Reis Santana (Universidad Estadual Paulista, Brasil)
    Title:  On a variant of Hilbert's 16th problem

    Abstract: We study the number of limit cycles that a planar polynomial vector field can have as a function of its number m of monomials. We prove that the number of limit cycles increases at least quadratically with m and we provide good lower bounds for mle10. Coauthored with Armengol Gasull.

  • February 19, 2024, 15:00
    Josep Maria Mondelo (Universitat Autònoma de Barcelona)
    Title:  The flow map parameterization method for invariant tori of autonomous Hamiltonian systems

    Abstract: The goal of this talk is to present a methodology for the computation of lower-dimensional (non-Lagrangian), partially hyperbolic invariant tori of autonomous Hamiltonian systems. It is therefore well suited (but not limited to) models from Celestial Mechanics such as the circular, spatial RTBP, spatial Hill, or the gravity field around a uniformly rotating, irregular body. It combines flow map (a.k.a. stroboscopic map) methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one, parameterization methods make the computational cost negligible with respect to (the unavoidable) numerical integration. Symplectic geometry is responsible for some cancellations that make the cohomological equations solvable. The method simultaneously computes the tori and their invariant stable and unstable bundles. An application is made to the computation of the Lissajous family of tori around $L_1$ in the Earth-Moon circular, spatial RTBP. In this application, when compared to more classical ("large-matrix") discretization techniques of the invariance equations, the speedup factor of the method is $>10$.

  • November 27, 2023, 15:00
    Yohanna Paulina Mancilla Martinez (Universidad del Bío-Bío)
    Title: Puiseux Series to Study the Liouvillian Integrability of a Special Quadratic System with Invariant Hyperbola (QSH) 

    Abstract: In this presentation, I will first introduce some definitions related to Liouvillian integrability and the Puiseux series method used to analyze the existence of Darboux polynomials associated with a planar polynomial dynamical system. The main objective of this work is to study the Liouvillian integrability of a family of QSH. Additionally, I will provide an upper bound on the degrees of irreducible invariant algebraic curves. This bound is essential for establishing the existence of Liouvillian first integrals. This research is a collaborative effort with Claudia Valls from Instituto Superior Técnico, Universidade Técnica de Lisboa, and the findings will be featured in the proceedings of the Royal Society of London. Series A.

  • October 16, 2023, 15:00
    Paulo Ricardo da Silva (IBILCE-UNESP-SJRP, Brazil)
    Title: Piecewise Smooth 1-Foliations.

    Abstract: We consider piecewise smooth 1-foliations generated by a pair of local smooth vector fields X, Y. The flow on the switching set depends on the regularization $Z_{\epsilon}$ which is a slow-fast system. Applying techniques of the geometric singular perturbation theory we study the  limit periodic sets  obtained as limit of orbits of  $Z_{\epsilon}$ when $\epsilon\rightarrow 0.$

  • May 29, 2023, 15:00
    Joyce Casimiro (Universidade Estadual de Campinas, Brazil)
    Title: Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds.

    Abstract: The Euler characteristic of a $2$-dimensional compact manifold and the local behavior of smooth vector fields defined on it are related to each other by means of the Poincaré-Hopf Theorem. Despite the importance of Filippov vector fields, concerning both their theoretical and applied aspects, until now, it was not known if this result is still true for Filippov vector fields. We show it is. While in the smooth case, the singularities consist of the points where the vector field vanishes, in the context of Filippov vector fields the notion of singularity also comprehends new kinds of points, namely, pseudo-equilibria and tangency points. Here, the classical index definition for singularities of smooth vector fields is extended to singularities of Filippov vector fields.  Such an extension is based on an invariance property under a regularization process. With this new index definition, we provide a version of the Poincaré-Hopf Theorem for Filippov vector fields. Consequently, we also get a Hairy Ball Theorem in this context, i.e. ``any Filippov vector field defined on a sphere must have at least one singularity (in the Filippov sense)''.

  • May 22, 2023, 15:00
    Àlex Arenas (Universitat Rovira i Virgili)
    Title: Unraveling the Complexities of Epidemic Spreading: Physics and Human Mobility Patterns.

    Abstract: Reaction-diffusion processes are fundamental to understanding epidemic dynamics and ecological systems in networked metapopulations. Within the context of epidemics, reaction processes signify contagion events within subpopulations, while diffusion denotes individual mobility between subpopulations. Recent research highlights the importance of recurrent human mobility patterns in explaining the transition to endemic epidemic states. In this study, we devise a comprehensive framework encompassing elementary epidemic processes, spatial population distribution, and commuting mobility patterns. Utilizing this framework, we distinguish three unique critical regimes of epidemic incidence based on these parameters. Intriguingly, one regime reveals a counterintuitive situation in which mobility inhibits disease propagation. We analytically determine the necessary conditions for the emergence of each of the three critical regimes in both real and synthetic networks. Additionally, we present a brief review of a tailored model for COVID-19 transmission in Spain, illustrating the practical application of our framework in deciphering real-world epidemic scenarios.

  • April 17, 2023, 15:00
    Marc Jorba-Cuscó (Centre de Recerca Matemàtica)
    Title: Jet transport and applications.
    Abstract: Variational equations for Ordinary Differential Equations (ODEs) are differential equations that describe how small changes to initial conditions or parameters affect the ODE solution. Calculating high-order derivatives of the vector field can be difficult, especially for high-dimensional systems. Automatic differentiation is a powerful computational technique that breaks down functions into smaller operations and applies the chain rule of differentiation to each operation. In this talk, we will explore the practical applications of Automatic differentiation in ODE solvers (jet transport). Jet transport allows for efficient computation of high-order derivatives of the ODE solution with respect to initial conditions or parameters. We will show that jet transport is algorithmically equivalent to applying the stepper to the whole set of variational equations, ensuring error is of the same order as the stepper. Additionally, we will discuss how jet transport impacts the choice of step size and provide various applications of the technique.
    Slides of the talk
  • March 13, 2023, 15:00
    Lucas Arakaki (Universidade Estadual Paulista, Brazil)
    Title: Detecting nilpotent centers on center manifolds of three-dimensional systems.
    Abstract: Consider analytical three-dimensional differential systems having a singular point at the origin such that its linear part is $y\partial_x-\lambda z\partial_z$ for some $\lambda\neq 0$. The restriction of such systems to a Center Manifold has a nilpotent singular point at the origin. Our goal is to study whether this singular point is a center on a center manifold or not without going through polynomial approximations. We study several tools to study these systems, among them, the formal integrability, the formal inverse Jacobi multiplier method, and approaching nilpotent centers as limits of Hopf-type centers. We use these results to solve the center problem for some three-dimensional systems without restricting the system to a parametrization of the center manifold.
    Slides of the talk
  • February 27, 2023, 15:00
    Nicolae Vulpe (Academy of Sciences, Moldova)
    Title: On the family QSL3 of quadratic systems with invariant lines of total multiplicity exactly 3.
    Abstract:  We consider the family QSL3 of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers, the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition, some subfamilies of quadratic systems possessing invariant lines of total multiplicity of at least three were also studied, among them the Lotka-Volterra family. However, there were still systems in QSL3  that remain to be studied. So we complete the study of the geometric configurations of invariant lines of QSL3 by studying all the remaining cases and give the full classification of this family modulo their configurations of invariant lines. This classification is done in affine invariant terms and we present here the most important invariant polynomials which are responsible for the existence and number of invariant lines for the family of quadratic systems. We also present the "bifurcation" diagram of the configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to QSL3  and in case it does, by producing its configuration of invariant straight lines. 

  • January 23, 2023, 15:00

    Claudio Pessoa (Universidade Estadual Paulista, Brazil)
    Title: Persistence of periodic orbits of rigid centers on two-dimensional center manifolds by quadratic perturbations.
    Abstract: We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centers on their center manifolds. Moreover, we obtain an example of a quadratic rigid center from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems.

  • September 19, 2022, 15:00

    Ronaldo A. Garcia (Universidade Federal de Goiás, Brazil)
    Title: Asymptotic lines of plane fields in three-dimensional manifolds. 
    Abstract: In this talk will be described the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of the hyperbolic region, where the Gaussian curvature is negative. When the plane field is completely integrable, these curves coincide with the classical asymptotic lines on surfaces. Joint work with Douglas Hilário.

  • July 18, 2022, 15:00

    Salomón Rebollo Perdomo (Universidad del Bío-Bío, Chile)
    Title: Dynamics of some nilpotent polynomial vector fields in $R^3$ 
    Abstract: We will present some results concerning the dynamics of a large family of nilpotent vector fields in $R^3$. The study is carried out from discrete and continuous points of view. In particular, we will show that some nilpotent vector fields have a surface foliated by periodic orbits.

  • July 11, 2022, 15:00

    Marc Chamberland (Grinnell College, EEUU)
    Title:  Newton's Method without Division
    Abstract: Newton's Method is the best-known iterative technique for root-finding. This quadratically converging technique has been applied to countless problems, including the basic arithmetic problems of calculating reciprocals and square roots, questions surrounding the relationships between various constants, and multi-trillion-digit calculations of Pi. A new variant of Newton's Method in one dimension has been found that avoids the division step while maintaining quadratic convergence. This talk will showcase the new method, experiments that lead to the main theorem, a proof that involves the dynamics of a complex rational function, and a peek into higher dimensions. The talk is aimed at a general audience.

  • July 4, 2022, 15:00
    Douglas Novaes (Universidade estadual de Campinas, Brazil)
    T    itle: Periodic solutions of Carathéodory differential equations via averaging method

    Abstract: In this talk, we approach the problem of the existence of periodic solutions for perturbative Carathéodory differential equations. The main result provides sufficient conditions on the averaged equation that guarantee the existence of periodic solutions. Additional conditions are also provided to ensure the uniform convergence of a periodic solution to a constant function. The proof of the main theorem is mainly based on an abstract continuation result for operator equations.
    Reference: An averaging result for periodic solutions of Carathéodory differential equations, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 150, Number 7, July 2022, Pages 2945–2954 https://doi.org/10.1090/proc/15810

  • May 9, 2022, 15:00
    Jordi Villadelprat (Universitat Rovira i Virgili)
    T    itle: On the cyclicity of Kolmogorov polycycles
    Abstract: In this talk, I will explain our recent results on the cyclicity of Kolmogorov polycycles. We study planar polynomial Kolmogorov's differential systems
    \[
     X_\mu\quad
       \left\{\!
       \begin{array}{l}
        \dot x=xf(x,y;\mu), \\[2pt] \dot y=yg(x,y;\mu),
       \end{array}
      \right.
    \]
    with the parameter $\mu$ varying in an open subset $\Lambda\subset\mathbb{R}^N$. Compactifying $X_\mu$ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle $\Gamma$, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all $\mu\in\Lambda.$ We are interested in the cyclicity of $\Gamma$ inside the family $\{X_\mu\}_{\mu\in\Lambda},$ i.e., the number of limit cycles that bifurcate from $\Gamma$ as we perturb $\mu.$ In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application, we study two cubic Kolmogorov families, with $N=3$ and $N=5$, and in both cases, we are able to determine the cyclicity of the polycycle for all $\mu\in\Lambda,$ including those parameters for which the return map along $\Gamma$ is the identity. In the talk, I will only state the general result and focus on the applications. This is a joint work with David Marín (UAB). 

  • March 14, 2022, 15:00
    Salvador Borrós (Universitat Autònoma de Barcelona)
    T    itle: Numerical Computation of Invariant Objects Using Daubechies Wavelets
    Abstract: 

    We present a method to compute the truncated wavelet expansion of an invariant object in a quasi-periodic skew product using periodic Daubechies Wavelets. To obtain the wavelet coefficients, we need to solve the invariance equations of the system in a mesh of points, desirably as dense as possible. This requires quite a bit of numerical finesse since Daubechies Wavelets do not have closed expressions. Once we have obtained the truncated wavelet expansion of the invariant object, we can compute its Besov regularity of said invariant objects. In particular, we will compute the regularity of Strange Non-Chaotic Attractors stemming from a Keller-GOPY biparametric family of systems as a proof of concept.
    (The talk will be in Catalan)