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

Coming talks

• March 13, 2023, 15:00
Lucas Queiroz Arakaki (Universidade Estadual Paulista, Brasil)
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

Past talks

• 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 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)
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)
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 the space. The study is carried out from the 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)
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 Douarte Novaes (Universidade estaduald e Campinas)
T
itle: Periodic solutions of Carathéodory differential equations via averaging method

Abstract: In this talk we approach the problem of 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 about 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)