Seminar on Topics in Dynamics
The 'Seminar on Topics in Dynamics' is a project of a recurrent seminar organized by Prof. Rodrigo D. Euzébio and co-organized by Prof. Durval Tonon and Prof. Luiz Gonçalves. It is an unfolding of the Research Project entitled Discontinuous singularly perturbed dynamical systems: theory and applications, funded by the National Council for Scientific and Technological Development (CNPq/Brasil - grant 402060/2022-9) - see the main page for more information on this project.
It is mentioned to be a regular seminar where local researchers and invited external speakers present short talks on general topics related to dynamical systems. But this is not all! An equally important goal is the communication that is supposed to be totally in English (not only the talk).
The talks occur every two weeks at the Auditorium of the Institute of Mathematics and Statistics of the Federal University of Goiás. You are certainly welcome to join us (we will provide a virtual room soon).
Schedule
Discontinuous singularly perturbed vector fields and beyond
Speaker: Prof. Dr. Rodrigo Donziete Euzébio - IME / UFG [CONFIRMED]
Day/time: October, 17th, 2023 - 11 a.m.
Abstract: In this talk, we are going to discuss some ideas connecting discontinuous vector fields and singularly perturbed ones. We will introduce some papers considering some bridges between those two objects, and we will also consider systems that natively present discontinuity and two (or more) time scales. A third ingredient will also be considered in our presentation, which is related to a type of non-smoothness on some specific boundaries. An example of a vector field achieving all of those features will be presented.
Título: Algebraic geometry and differential equations with interval coefficients
Speaker: Prof. Dr. Alain Jacquemard - Université de Bourgogne [TO BE CONFIRMED]
Day/time: October, 31th, 2023 - 10 a.m. [1/2]
Abstract: We discuss some singularities of differential equations with interval coefficients by means of algebraic varieties. We then compute invariants of such systems by building vector fields on regions of interest according to a natural stratification. Joint work with Marina T. Mizukoshi and Weldon A Lodwick
Título: On the minimal number of periodic orbits for nonsingular Morse-Smale flows
Speaker: Prof. Dr. Gioia Vago- Université de Bourgogne [TO BE CONFIRMED]
Day/time: October, 31th, 2023 - 11 a.m. [2/2]
Abstract: We consider the couples $(M, \Phi)$ where $M$ is an odd-dimensional compact manifolds with boundary, endowed with a non-singular Morse-Smale flow~$\Phi$, satisfying some \emph{given} homological boundary information. We compute, in terms of such given homological information, a number $p_{min}$ such that \emph{any} non-singular Morse-Smale flow~$\Phi$ on any manifold~$M$ satisfying the given abstract homological data must have \emph{at least} $p_{min}$ closed periodic orbits. Moreover, we can provide, for any initial homological data, a Morse-Smale model $(M_0, \Phi_0)$ for which $p_{min}$ is attained. Note that in the general case of a couple $(M, \Phi)$ satisfying the given homological information, such a number $p_{min}$ is a lower bound.
Ref.: M.A. BERTOLIM, C. BONATTI, M. MELLO, G.M. VAGO, Minimal number of periodic orbits for nonsingular Morse-Smale flows in odd dimension.
Title: Global dynamics of Planar Piecewise Linear Inelastic Systems Having Straight Lines as Switching Manifolds
Speaker: Yovani Villanueva - IME/UFG [CONFIRMED]
Day/time: November, 28th, 2023
Abstract: In this lecture we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider.
The mathematical work of J.Sotomayor: bifurcation theory and qualitative theory of principal lines on surfaces
Speaker: Ronaldo Alves Garcia - IME / UFG (Goiânia/GO) [CONFIRMED]
Day/time: December, 12th, 2023 - 11 a.m.
Abstract: In this thesis, the main topic is the classification of bifurcations appearing in generic one-parameter families of vector fields They were characterized in terms of their stability and regularity properties. Together with Carlos Gutierrez (in memoriam), they developed the qualitative theory of principal lines on surfaces. The main goal of this talk is to recall the main mathematical contributions of Jorge Sotomayor (1942-2022).
Fenichel theorem
Speaker: Prof. Luiz Fernando Gonçalves - IME / UFG [TO BE CONFIRMED]
Day/time: TBA
Abstract: TBA