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

**The GSDUAB Working Seminar is closed until September.**

**Past talks**

**May 29, 2023, 15:00****Joyce Casimiro**(Universidade Estadual de Campinas, Brazil)**T****itle:**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)**T****itle:**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)**T****itle:**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)**T****itle:****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)**T**On the family QSL3 of quadratic systems with invariant lines of total multiplicity exactly 3.**itle:****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)**T****itle:**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)**T****itle:**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)**T****itle:**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)**T****itle:****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**(Universidade estadual de Campinas, Brazil)*Douglas Novaes*

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**(Universitat Rovira i Virgili)*Jordi Villadelprat*

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**(Universitat Autònoma de Barcelona)*Salvador Borrós*

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)