This session will focus on the study of different aspects of dynamical systems, both in their discrete and continuous aspects, and from both a theoretical and applied point of view.
The topics covered will include global stability, attractors, chaos, bifurcations, classification problems and modeling of natural phenomena.
The session will provide a broad meeting environment for researchers in this area of mathematics, together with the related parallel session MA03 Nonlinear Dynamics and Applications.
Esta sesión tratará sobre el estudio de diferentes aspectos de los sistemas dinámicos, tanto en su vertiente discreta como continua, y tanto desde un punto de vista teórico como aplicado.
Entre las temáticas que se tratarán se incluyen cuestiones sobre estabilidad global, atractores, caos, bifurcaciones, problemas de clasificación y modelización de fenómenos naturales.
La sesión proporcionará un entorno amplio de encuentro a investigadores/as de esta área de las matemáticas, junto con la sesión paralela afín MA03 Dinámica no lineal y aplicaciones.
37-XX
(primary)
1.A (0.27)
1.B (0.27)
1.C (0.27)
We know that a real function defined on $[a,b]$ such that $f(a)f(b)<0$ and whose derivative never vanishes has exactly one zero in $[a,b]$. Nevertheless, it is less known that similar results hold for higher dimensions. In this talk we will expose a new result in this direction, based on a combination of the Poincaré-Miranda theorem and a mix of some old and new arguments for the injectivity issue. Finally, we will apply the criterion to an example and we will discuss some applications.
Joint work with Sebastián Buedo Fernández.
We investigate the dynamics near a normally elliptic invariant curve in a 3D volume-preserving map, reducing the map to a resonant Birkhoff normal form around the curve. This depends on the set of resonances between the tangent and normal frequencies to the curve. Single-resonances may destroy the curve, eventually leading to a chain of stability bubbles. When the elliptic curve persists, the normal dynamics become reducible, allowing the classification of 3D resonant structures around it.
Joint work with Arturo Vieiro.
Dissipative semigroups produce structures that are invariants and attracts every trajectory of the phase space, well know as global attractors, that are bounded. We are going to introduce the concept of an attractor that it is unbounded, the maximal attractor, and study their existence and properties, such as characterize it. Finally, we apply our result to a parabolic semilinear PDE, where the nonlinearity can be unbounded, as long as it grows linearly with a controlled growth constant.
Joint work with Matheus Bortolan, Juliana Fernandes and Piotr Kalita.
In this talk, we explore a two-parameter family of 3D diffeomorphisms related to a generic unfolding of a unipotent fixed point. We begin presenting a parameter subset with an interesting dynamical behaviour consisting of two saddle-focus fixed points with different unstable indices. After that, we examine the distances between the one-dimensional manifolds and explain how these dynamics may give rise to Tatjer's homoclinic tangencies.
Nonautonomous saddle-node bifurcations have often been studied under the condition of concavity of the flow; in previous works, we explored them under $d$-concavity properties. This talk weakens that condition, identifying such bifurcations in equations with d-concavity properties in measure. The new framework allows equation coefficients to vary within large chaotic sets, in some way approaching a stochastic formulation. Applications in circuit theory and critical transitions are also presented.
Joint work with Carmen Núñez and Rafael Obaya.
In this talk, we will start characterizing the Conley-Zehnder index in terms of the winding number of a linear planar periodic Hamiltonian system. This will allow us to apply the Poincaré-Birkhoff theorem in order to prove existence and multiplicity of periodic solutions in general nonlinear and nonautonomous Hamiltonian systems. Finally, some aplications will be provided.
Joint work with Alberto Boscaggin.
This talk is meant to be a friendly introduction to complex dynamics, starting from the iteration of $z^2$, and providing the definition of the basic concepts in the field: the Fatou set (stability) and the Julia set (chaos). The goal is to describe the dynamics on the boundaries of Fatou components (connected components of the Fatou set), from a measure-theoretical, symbolic and qualitative point of view, in the case when the iterated function is a transcendental entire function.
Joint work with Núria Fagella.
The Takagi function is a classical example of a continuous nowhere differentiable function. It is defined as $$T(x)=\sum_{n=0}^{\infty}\frac{\phi(2^n x)}{2^n},\quad x\in [0,1]$$where $\phi(x)$ denotes the distance from the point $x$ to the nearest integer. In this talk, we will study the discrete dynamical system generated by the Takagi function, namely $$x_{n+1}=T(x_n),\quad x_0\in [0,1].$$
Joint work with Zoltán Buczolich.
The study of neural populations is of increasing interest. In the literature, there are two mean-field models representing the dynamics of heterogeneous all-to-all networks of QIF neurons with and without synaptic dynamics. In this presentation, we study the different dynamical changes observed when a parameter (linking both models and related with the synapsis) is varied, and we analyze the bifurcations underlying these changes.
Joint work with R. Barrio, J.A. Jover-Galtier, C. Mayora-Cebollero, S. Serrano and L. Pérez.
In the studies developed with E. Liz and F. M. Hilker, we considered discrete 1D population models subject to control rules that combine constant quota and threshold harvesting. These combination lead in a natural way to piecewise-smooth maps whose dynamics are challenging because multiple non-smooth bifurcations may appear. The main aim of this talk is to show how we have procced to determine the asymptotic dynamics of the models by studying the “geometry” of the associated non-smooth maps.
Joint work with E. Liz and F. M. Hilker.
The chaoticity analysis of a dynamical system is usually performed with classical techniques as Lyapunov Exponents. Recently, Deep Learning (DL) has also been used to obtain such analysis. However, when working with real data, classical and DL techniques have drawbacks. In this presentation, we show how DL can be used to obtain the chaoticity analysis of theoretical data (3D analysis of Lorenz system), and we propose a DL chaoticity algorithm for the analysis of real data (frog heart dynamics).
Joint work with R. Barrio, F.H. Fenton, Á. Lozano, A. Mayora-Cebollero, A. Miguel, A. Ortega, S. Serrano and R. Vigara.