Seminário de Sistemas Dinâmicos 2025/1
SEMINÁRIOS DE SISTEMAS DINÂMICOS - 2025/1
Coordenadora dos seminários: Profa. Kamila da Silva Andrade
Todos os seminários serão transmitidos através do link: https://meet.google.com/msh-pbyc-pkr
Seminários
Programação
Seminário 1: 18/03/2025
Título do seminário: Generic Bifurcation of Symmetric Two-Zone Piecewise Vector Fields on R³
Ministrante: Prof. Dr. João Carlos da Rocha Medrado - IBILCE/UNESP
Data: 18/03/2025
Horário: 10h
Local: Pelo link acima
Resumo: Following Smale's Program, near symmetric singularities, we characterize the set of generic bifurcation of symmetric two-zone piecewise vector fields on R³. We give the intrinsic conditions and normal forms of these vector fields such that they are of codimensions zero and one.
Joint work with U. Castro.
Seminário 2: 25/03/2025
Título do seminário: Piecewise Smooth Vector Fields where the Switching Manifold is a Double Discontinuous
Ministrante: Dr. Mayk Joaquim dos Santos - Egresso PPGMAT-IME-UFG
Data: 25/03/2025
Horário: 10h
Local: Auditório do IME e pelo link acima
Resumo: At the seminary, main purpose exhibit the open and dense subset of piecewise smooth vector fields that is structural stable in 2D, following the Thom-Smale's program, where the switching manifold is a double discontinuity and therefore the Filippov's convention is not applied.
Seminário 3: 01/04/2025
Título do seminário: Perturbing periodic integral manifold of non-smooth differential systems
Ministrante: Dr. Oscar Alexander Ramírez Cespedes - Universidad Distrital Franciso José de Caldas - Bogotá/Colômbia
Data: 01/04/2025
Horário: 10h
Local: Pelo link acima
Resumo: This talk addresses the perturbation of higher-dimensional non-smooth autonomous differential systems characterized by two zones separated by a codimension-one manifold, with an integral manifold foliated by crossing periodic solutions. Our primary focus is on developing the Melnikov method to analyze the emergence of limit cycles originating from the periodic integral manifold. While previous studies have explored the Melnikov method for autonomous perturbations of non-smooth differential systems with a linear switching manifold and with a periodic integral manifold, either open or of codimension 1, our work extends to non-smooth differential systems with a non-linear switching manifold and more general periodic integral manifolds, where the persistence of periodic orbits is of interest. We illustrate our findings through several examples, highlighting the applicability and significance of our main result.
Seminário 4: 08/04/2025
Título do seminário: Generic upper bounds of cyclicity problem
Ministrante 2: Dr. Yovani Villanueva - IME/UFG
Data: 08/04/2025
Horário: 9h30
Local: Auditório do IME e pelo link acima
Resumo: Hilbert’s 16th problem remains without a definitive solution, even for quadratic systems, and the center-focus problem has also proven elusive for polynomial vector fields of degree $n \geq 3$. In this talk, I provide novel results to get a local analytic first integral of a singularity at the origin, using Lyapunov formula and computer-assisted tools, for generic polynomial differential systems with center linearization in $\mathbb{R}^2$. From those results, upper bounds for the maximum number of center conditions and small limit cycles are obtained, in polynomial vector fields of any finite degree at the origin.
Título do seminário: Limit cycles for piecewise rigid systems with homogeneous non-linearities
Ministrante 1: Dr. Joan Torregrosa - Universitat Autònoma de Barcelona
Data: 08/04/2025
Horário: 10h30
Local: Auditório do IME e pelo link acima
Resumo: The study of limit cycles in piecewise systems can be approached by analyzing the composition of the return map in each region. In fact, this problem can be reinterpreted as the study of the fixed points of the composition of k distinct diffeomorphisms. However, in general, this problem is quite challenging, often intractable, because the diffeomorphisms associated with the return maps are not explicitly known. To make progress, we will consider a special case where all k diffeomorphisms have explicit forms. Our primary focus will be to determine the maximum number of fixed points and investigate their stability. These fixed points can also be viewed as the zeros of the difference map. In this talk, we will present results from both local and global perspectives. Specifically, we will demonstrate how the problem can be studied in the perturbative near-identity case, focusing on the search for higher multiplicity zeros and their unfoldings. We will employ techniques from the qualitative theory of differential equations that are particularly useful in this context.
The talk is based in a current joint work with Armengol Gasull.
Seminário 5: 15/04/2025
Título do seminário: Conley’s theory in the study of Filippov vector fields
Ministrante 2: Ma. Letícia Cândido - IMECC/Unicamp
Data: 15/04/2025
Horário: 10h00
Local: Pelo link acima
Resumo: A Teoria de Conley é uma poderosa ferramenta na análise qualitativa de sistemas dinâmicos, oferecendo uma estrutura topológica baseada no conceito de conjuntos invariantes isolados e nos chamados índices de Conley. Esses conceitos permitem uma compreensão mais profunda da estrutura global do espaço de fases, mesmo na ausência de soluções explícitas.
Nos últimos anos, tem havido um crescente interesse na aplicação da Teoria de Conley a sistemas não suaves, como os campos de Filippov, que descrevem sistemas dinâmicos com descontinuidades. Esses campos surgem naturalmente em diversas áreas, como controle, eletrônica, economia e mecânica com impacto.
A adaptação da Teoria de Conley para campos de Filippov envolve o desenvolvimento de ferramentas conceituais e computacionais que levem em conta o comportamento nas superfícies de descontinuidade. Isso inclui a extensão do conceito de fluxo semifluxo, a definição adequada de conjuntos invariantes e a construção de índices que reflitam a complexidade da dinâmica na região de descontinuidade.
O estudo dessa interação entre topologia e sistemas não suaves contribui para o avanço teórico e prático na análise de sistemas dinâmicos com descontinuidades.
Seminário 6: 22/04/2025
Título do seminário: Results about structural stability and the existence of limit cycles for piecewise smooth linear differential equations separated by the unit circle
Ministrante 2: Dra. Mayara Duarte de Araujo Caldas - UFRJ
Data: 22/04/2025
Horário: 10h00
Local: Pelo link acima
Resumo: In this talk, we investigate the structural stability and the existence of limit cycles in families of piecewise smooth differential equations where the unit circle serves as the discontinuity region. Our study encompasses families featuring singularities of center or saddle type, both visible and invisible, as well as those without any singularities. For the family that admits only constant vector fields, we describe the dynamics over and present a result regarding structural stability. For the other families, we provide an upper bound for the number of limit cycles and present examples that illustrate the maximum number of limit cycles that can be realized.
Seminário 7: 29/04/2025
Título do seminário: Hidden Dynamics: resolving singularities and raising new mysteries
Ministrante 2: Dr. Mike Jeffrey - University of Bristol
Data: 29/04/2025
Horário: 10h00
Local: Pelo link acima
Resumo: Discontinuities are inescapable in modelling dynamical systems, particularly in engineering and biology, anywhere afflicted by switches, decisions, impacts, or cell division. Despite a century of work developing the theory of these “nonsmooth” dynamics systems, two big challenges remain: indeterminacy (equations are non-unique at a discontinuity), and a curse of dimensionality (every new dimension brings new classification problems, so there can be no general theory in n-dimensions as we have for smooth systems).
I will describe how ‘hidden dynamics’ partially resolves this, allowing us to study nonsmooth systems using methods from smooth theory, trying to remove the curse of dimensionality. Also, whereas many current theoretical works try to banish indeterminacy from nonsmooth dynamics, hidden dynamics shows that it is essential, as it also allows us to resolve some singularities, while revealing others that render systems highly unpredictable. I’ll introduce a singularity that challenges our ideas of structural stability of nonsmooth systems, and might hide in the mathematics of the decisions we make every day.
Seminário 8: 06/05/2025
Título do seminário: Puiseux inverse integrating factors and Puiseux first integrals at monodromic singularities
Ministrante 2: Dra. Ana Livia Rodero - ICMC Usp
Data: 06/05/2025
Horário: 10h00
Local: Pelo link acima
Resumo: The main goal of this talk is to show that neither a Puiseux first integral H nor a Puiseux inverse integrating factor V can characterize degenerate centers in planar vector fields with a monodromic singularity. We also show that the existence of a Puiseux first integral H is a sufficient center condition.
It is part of a joint work with Prof. Isaac A. García and Prof. Jaume Giné, from Universitat de Lleida (UdL/Spain).
[1] García IA, Giné J, Rodero AL, Existence and nonexistence of Puiseux inverse integrating factors in analytic monodromic singularities, Stud Appl Math. 2024;153:e12724. https://doi.org/10.1111 /sapm.12724
Seminário 9: 20/05/2025
Título do seminário: Dynamics at the boundary of planar regions
Ministrante: Mr. Adimar Moretti Júnior - IME/UFG
Data: 20/05/2025
Horário: 10h00
Local: Auditório do IME e pelo link acima
Resumo: In this lecture, we propose a new approach to deal with the dynamics at the boundary of some planar regions in which linear or non-linear vector fields are defined. In particular, details about a linear case defined on a circle will be discussed, as well as future work to be developed.
Seminário 10: 27/05/2025
Título do seminário : On the Bifurcation of a Typical Homoclinic-like Loop from a 3D Cusp-Fold Singularity
Ministrante: Dr. Otávio Gomide - IME/UFG
Data: 27/05/2025
Horário: 10h00
Local: Link acima
Resumo: Generic Bifurcation Theory is a well-established area of Dynamical Systems which has been extensively studied in the last years due to its importance in providing a comprehensive understanding of both applied and theoretical problems. In light of this, extending this area to address the new dynamics arising from the recent Theory of Filippov systems becomes mandatory in order to enlarge the knowledge on phenomena involving discontinuous motion.
In this context, we study the local bifurcations of a 3D Filippov system around a cusp-fold singularity. We detect that such kind of point may exhibit some degeneracies which give rise to the local bifurcation of global connections between Σ-singularities. More specifically, we prove that a homoclinic-like loop at a fold-regular singularity bifurcates from a codimension 2 cusp-fold singularity.
This is a joint work with O. Cespedes and R. Cristiano.
Seminário 11: 03/06/2025
Título do seminário: Campos Vetoriais Próximos a Subvariedades
Ministrante: Dr. Pablo Vandré Jacob Furlan - IFG
Data: 03/06/2025
Horário: 10h00
Local: Link acima
Resumo: Neste seminário apresentarei os resultados de Ishikawa, Izumiya e Watanabe, que resolveram um dos problemas propostos por V. I. Arnold em seu artigo On Local Problems of Analysis e, com isso, generalizaram uma das formas normais de Vishik sobre campos vetoriais próximos a subvariedades. Em seguida, introduzirei a fórmula de índice que desenvolvi e mostrarei sua aplicação em campos vetoriais quadráticos descontínuos. Por fim, discutirei como essa mesma abordagem pode ser estendida a campos contínuos para analisar configurações de singularidades e órbitas periódicas, bem como para definir índices em grafos de órbitas heteroclínicas.
Seminário 12: 10/06/2025
Título do seminário 1: On Weak Foci and Limit Cycles in Planar Cubic Differential Systems
Ministrante 1: Mr. Gerardo Anacona Erazo - IME/UFG
Data: 10/05/2025
Horário: 10h00
Local: Auditório do IME e pelo link acima
Resumo 1: A classical and challenging problem in the qualitative theory of planar differential systems is that of distinguishing between a center and a focus
In this presentation, we characterize the conditions under which the origin is a weak
focus of the following cubic polynomial differential system:
x ̇ = −y, y ̇ = x + a1x²+ a2xy + a3y²+ A,
where A is an arbitrary nonzero monomial of degree 3. Furthermore, we identify limit
cycles bifurcating from this weak focus and provide representative phase portraits in the
particular case A = a4x³.
Título do seminário 2: Limit cycles bifurcating from an invisible two-fold singularity in planar piecewise - smooth generalized Liénard systems
Ministrante 1: Ma. Marly Tatiana Anacona Cabrera - IME/UFG
Data: 10/05/2025
Horário: 11h00
Local: Auditório do IME e pelo link acima
Resumo 2: In this work, we analyze a planar piecewise smooth generalized polynomial Liénard system with a switching boundary at x = 0. We first determine conditions on the system parameters that ensure the existence of an invisible two-fold singularity, characterized by quadratic contact of both vector fields with the switching boundary. Then, we provide a detailed construction of the associated Poincaré map in a neighborhood of the two-fold, and we derive conditions under which: (i) two types of Hopf-like bifurcations occur, supercritical and subcritical, from which a crossing limit cycle (CLC) bifurcates from the two-fold, being stable in the supercritical case and unstable in the subcritical case; (ii) a saddle-node bifurcation of CLCs occurs, where two CLCs born from the two-fold collide and then vanish. The local behavior around the two-fold point is summarized in a two-parameter bifurcation set having a bifurcation point of codimension 2 from which three branches of bifurcation curves of codimension 1 emanate, two of which are related to Hopf-like bifurcations and one is related to the saddle-node bifurcation of CLCs.
Seminário 13: 17/06/2025
Título do seminário 1: Slow-Fast Filippov Sliding Vector Field
Ministrante 1: Ma. Vitória Chaves Fernandes - IME/UFG
Data: 17/05/2025
Horário: 10h00
Local: Auditório do IME e pelo link acima
Resumo 1: In this presentation, we consider a system defined by two linear vector elds in R³, separated by a switching manifold, Σ. Each vector eld is a slow-fast system. As a consequence of the discontinuity at Σ, the critical manifolds are distinct. We present results that connect these two slow fast systems and prove that, under certain conditions, the sliding vector field also exhibits the behavior of a slow-fast system. Furthermore, we apply the obtained results to a simple class of model problems corresponding to single-input-single-output, linear, time-invariant relay control systems with unit negative feedback of the output variable.
References
[1] Di Bernardo, M., Johansson, K.H., Vasca, F.: Self-oscillations and sliding in re-
lay feedback systems: symmetry and bifurcations. Int. J. Bifurc. Chaos 11(04), 11211140(2001).
[2] Euzébio, R.D., Fernandes, V.C.: Three-dimensional fast-slow Filippov systems. Preprint(2025).
[3] Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations.Journal of Differential Equations, 31(1), 53-98 (1979).
[4] Filippov, A.F.:, Differential equations with discontinuous righthand sides, volume 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian (1988).
Título do seminário 2: Limit cycles of discontinuous piecewise differential systems formed by two rigid systems separated by a straight line
Ministrante 1: Ma. Angela Carolina Tunubalá Sánchez - IME/UFG
Data: 17/05/2025
Horário: 11h00
Local: Auditório do IME e pelo link acima
Resumo 2: In this work, we study the maximum number of limit cycles in some classes of discontinuous piecewise differential systems formed by two rigid systems separated by a straight line. These rigid systems consist of a linear center plus a homogeneous polynomial of degree 2, 3, 4, 5, or 6. The famous 16th Hilbert problem asks for an upper bound on the maximum number of limit cycles that planar polynomial vector fields of a given degree can exhibit, see [2]; to date, this problem remains open. For these classes of piecewise differential
systems, we solve the extended 16th Hilbert problem.
References
[1] Llibre, J., Sanchez, A. C., and Tonon, D. J , Limit cycles of discontinuous piecewise differential systems formed by two rigid systems separated by a straight line. Nonlinear Differential Equations and Applications NoDEA, 32(3), 48, (2025).
[2] Hilbert, D., Mathematische Probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. G”ottingen Math. Phys. KL. (1900), 253–297; English transl., Bull. Amer.
Math. Soc. 8 (1902), 437–479; Bull. (New Series) Amer. Math.Soc. 37 (2000), 407–436.
Seminário 14: 24/06/2025
Título do seminário 1: Sewing Limit Cycles in Discontinuous Piecewise Smooth Vector Fields with Two Linear Centers and a Coupled Rigid Center
Ministrante 1: Mr. Warley Mendes Batista - IME/UFG
Data: 24/05/2025
Horário: 10h00
Local: Auditório do IME e pelo link acima
Resumo 1: In this seminar we present an upper bound for the maximum number of limit cycles that can arise in piecewise vector fields formed by centers, defined across three zones separated by two parallel lines. In this setup the smooth vector field on the left exhibits an arbitrary linear center, the central vector field has another arbitrary linear center, and the vector field on the right has a rigid center. We study two cases: when the rigid center ̇x = −y + xP(x, y), y ̇ = x + yP(x, y), has P(x, y) is a polynomial of degree one, and when it is a of degree two. In the first case the maximum number of limit cycles is seven and there are systems with three limit cycles. In the second case the maximum number of limit cycles is four, and this number is achieved.
Seminário 15: 01/07/2025
Título do seminário 1: Bifurcações de Hopf em sistemas planares de Equações Diferenciais
Ministrante 1: Ma. Thaylline Rocha Madureira - IME/UFG
Data: 01/07/2025
Horário: 10h00
Local: Auditório do IME e pelo link acima
Resumo 1: Nesta apresentação, analisamos sistemas bidimensionais de equações diferenciais da forma x′ = f(x, α), onde α é um parâmetro real e x um vetor bidimensional. Exploramos as condições para o surgimento de uma bifurcação de Hopf, discutimos sua forma normal e ilustramos seu comportamento por meio de exemplos no modelo predador-presa.
Referências
[1] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York
(2004).
[2] Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn.
Springer, New York (2003).
[3] Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York
(2001).
[4] Zhou, Y., Sun, W., Song, Y., Zheng, Z., Lü, J., Chen, S.: Hopf bifurcation analysis of a
predator–prey model with Holling-II type functional response and a prey refuge. Nonlinear
Dynamics 97(2), 1439–1450 (2019). Springer.
Título do seminário 2: Ciclos Limites no Toro de Dupin Perturbado
Ministrante 1: Ma. Deysquele do Nascimento Ávila - IME/UFG
Data: 01/07/2025
Horário: 11h00
Local: Auditório do IME e pelo link acima
Resumo 2: Análogo ao 16º problema de Hilbert, este trabalho investiga a existência e o comportamento de ciclos limite definidos sobre o toro de Dupin, uma superfície clássica da geometria diferencial. A motivação central é determinar o número máximo de órbitas fechadas isoladas, adaptando essa perspectiva ao contexto topológico do toro. Para isso, analisamos equações diferenciais binárias da própria geometria da superfície. O estudo integra técnicas da teoria qualitativa das equações diferenciais, da geometria diferencial e da teoria da média, com ênfase no comportamento das linhas de curvatura e linhas assintóticas do toro.
Seminários anteriores:
Seminários de Sistemas Dinâmicos 2024-2
Seminários de Sistemas Dinâmicos 2024-1
Seminários de Sistemas Dinâmicos 2023-2
Seminários de Sistemas Dinâmicos 2023-1
Seminários de Sistemas Dinâmicos 2022-2
Seminários de Sistemas Dinâmicos 2022-1
Seminários de Sistemas Dinâmicos 2021-2
Seminários de Sistemas Dinâmicos 2020-2 e 2021-1.
Seminários de Sistemas Dinâmicos 2020-1.
Seminários de Sistemas Dinâmicos 2019.
Seminários de Sistemas Dinâmicos 2018.
Seminários de Sistemas Dinâmicos 2017.
Seminários de Sistemas Dinâmicos 2016.
Seminários de Sistemas Dinâmicos 2015.
Seminários de Sistemas Dinâmicos 2014.