Dia: Dimecres, 27 de setembre de 2017
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Tomás Lázaro, Universitat Politécnica de Catalunya
Títol: At the origins of life: small hypercycles with short-circuits
Resum: One of the most challenging open problems in science is to understand the origins of life in Earth primitive state and, in particular, concerning the first primitive replicating systems. Among the existent theories appearing in the last decades, the one conceived by Manfred Eigen and Peter Schuster (1971,1979) states that so-called hypercycles could play a crucial role since they could explain the growth in complexity overcoming the error threshold or critical mutation rate phenomenon. Hypercycles, i.e. nonlinear catalytic networks, allow an all-species coexistence and could support an information content larger than the one found for a quasispecies-based model. This theory received important criticisms due to its high sensitivity to the so-called parasites and short-circuits. While the impact of parasites has been widely investigated for well-mixed and spatial hypercycles, the effect of short-circuits in hypercycles remains poorly understood. In this talk we will present, briefly, a dynamical description of two small asymmetric hypercycles with short-circuits, tackling the question of growing complexity while keeping some stability.
This is a joint work with Ernest Fontich (UB), Toni Guillamon (UPC) and Josep Sardanyés (CRM).
Dia: Dimecres, 18 d'octubre de 2017
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Rodrigo G. Schaefer, Universitat Politécnica de Catalunya
Títol: Scattering maps and global instability in Hamiltonian systems
Resum: In this work we illustrate the Arnold diffusion in a concrete example: the a priori unstable Hamiltonian system of two and a half degrees of freedom:
H(p,q,𝑰,φ,s) =½p² + cos q -1 +½𝑰² + h(q,φ,s;ε),
proving that for any small periodic perturbation of the form
h(q,φ,s;ε) = εcos q (a₁cos(k φ + ls) + a₂cos (k'φ + l's ) )
(a₁a₂ ≠ 0, k l'≠ k' l, and ε ≠ 0 small enough) there is global instability for the action 𝑰. For this, we apply a geometrical mechanism based in the explicit computation of several scattering maps.
This is a joint work with Amadeu Delshams.
Dia: Dimecres, 25 d'octubre de 2017
Lloc: Aula IMUB (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Sergey Gonchenko, Institute for Applied Mathematics and Cybernetics, N. Novgorod, Russia
Títol: Shilnikov heritage: review of his classical and pioneering works
Resum: I will try to fulfil an impossible task: to give a more or less intelligible overview of the pioneer classical L.P.Shilnikov works of 60-80th years. The main Shilnikov results obtained in this period can be grouped into 5 main topics:
A càrrec de: Dmitry Turaev, Imperial College, UK.
Títol: A positive metric entropy conjecture
Resum: We prove that any area-preserving diffeomorphism with an elliptic periodic point can be perturbed, in C-infinity topology, to one exhibiting a chaotic island with positive metric entropy (a joint work with P.Berger)
Dia: Dimecres, 22 de novembre de 2017
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Pau Martín, Universitat Politècnica de Catalunya
Títol: Invariant manifolds of Diophantine parabolic tori and where to find them
Resum: What are Diophantine parabolic tori? Do these fantastic beasts really exist? Where do they live? Do they have whiskers? Are they of any interest? In this talk we'll try to answer these and other questions.
Dia: Dimecres, 29 de novembre de 2017
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Pere Gutiérrez, Universitat Politècnica de Catalunya
Títol: Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
Resum: The splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a 3-dimensional torus with a fast frequency vector ω/√ε, with ω=(1,Ω,Ω²) where Ω is a cubic irrational number whose two conjugates are complex (for instance, the real root of z³+z-1). Applying the Poincaré-Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov function. This allows us to provide an asymptotic estimate for the maximal splitting distance, which is exponentially small in ε, and valid for all sufficiently small values of ε. The function in the exponent turns out to be quasiperiodic with respect to log(ε), and can be explicitly constructed from the resonance properties the frequency vector $\omega$. In this way, we emphasize the strong dependence of our results on the arithmetic properties of the frequencies. This is a joint work with Amadeu Delshams and Marina Gonchenko.
Dia: Dimecres, 13 de desembre de 2017
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Juan José Morales, Universitat Politécnica de Madrid
Títol: Teoría de Galois diferencial y no integrabilidad de campos polinomiales en el plano
Resum: En esta charla estudiaremos una condición necesaria de integrabilidad de campos polinomiales en el plano, mediante teoría de Galois diferencial. Más concretamente, como un corolario de un resultado previo con Ramis y Simó sobre sistemas Hamiltonianos, se ha probado que una condición nesesaria para la existencia de una integral primera meromorfa es que la componente de la identidad de las ecuaciones en variaciones de orden arbitrario alrededor de una solución, debe ser abeliano. Mediante un teorema de Liouville, el problema es equivalente a la existencia de una solución racional de una cierta ecuación lineal de primer orden, la ecuación de Risch. Este es un problema clásico estudiado por Risch en 1969, y la solución viene dada por el llamado algoritmo de Risch. Tambien trataremos de aclarar la conexión de nuestro trabajo con ciertas funciones transcendentes no elementales, como la función error. Este es un trabajo conjunto con P. B. Acosta-Humánez, J.T. Lázaro y C. Pantazi.
Dia: Dimecres, 20 de desembre de 2017
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Toni Guillamon, Universitat Politécnica de Catalunya
Títol: Mathematical strategies to estimate synaptic conductances
Resum: We will present different approaches to estimate synaptic conductances and discern between excitatory and inhibitory inputs using only the information of the time course of the membrane potential of a neuron. In neuroscience, this information can provide insights on the excitation/inhibition balance in parts of the brain. From a mathematical point of view, the purpose of this problem is to quantify the input that a multidimensional externally forced system is receiving from the only information of a single variable. It results, then, in an inverse problem and, as other estimation problems, it brings up interesting challenges for many branches of mathematics: statistics, stochastic processes, dynamical systems, Bayesian inference, optimization,... In this talk, we will focus on aspects closer to dynamical systems, but we will also try to give an overview of other methods. In particular, we will discuss on the influence of active ionic channels, and the necessity of both inferring conductances from single trials and providing model-free methods. We will show our proposals to overcome nonlinear effects in the subthreshold regime, as well as a proof of concept for the spiking regime. The contents of this lecture are based on different collaborations with Rune W. Berg, Pau Closas, Susanne Ditlevsen, Rafel Prohens, Antonio E. Teruel and Cati Vich.
Dia: Dimecres, 14 de febrer de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Joan Torregrosa, Universitat Autònoma de Barcelona
Títol: Hilbert numbers using reversible centers
Resum: We will show the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree N,H(N), for small values of N. These limit cycles appear bifurcating from new symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric about it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least 2n+m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, H(4)≥ 28, H(5)≥ 37, H(6)≥ 53, H(7)≥ 74, H(8)≥ 96, H(9)≥ 120, and H(10)≥ 142.
The talk is based in a joint work with Rafel Prohens. First I will do a short review about the degenerated Hopf bifurcation (Liapunov quantities). In particular how we can use a cluster of computers for the parallelization computation of all the necessary conditions (local integrability conditions) to study the local unfoldings that guaranty the existence of the small limit cycles.
R. Prohens, J. Torregrosa. New lower bounds for the Hilbert numbers using reversible centers. Preprint. 2017
Dia: Dimecres, 21 de febrer de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Cezary Olszowiec, Imperial College
Títol: Shadowing of the non-transverse heteroclinic chains: an example
Resum: We consider a problem of shadowing of a network in ℝ⁴ consisting of hyperbolic equilibria and non-transverse heteroclinic connections. Such a heteroclinic network naturally arises from a simple toy-model of a cyclic competition bimatrix game (Rock-Scissors-Paper) governed by the coupled replicator equations depending on two parameters.
During the talk, we will recall some fundamental results on asymptotic behaviour and infinite switching in the neighbourhood of a heteroclinic network. We will also mention results relating the considered problem to the other models from Game Theory (i.e. Best Response Dynamics). We will pose some questions and present our investigations on shadowing, existence of superheteroclinic orbits and global bifurcations occurring in the considered system.
Dia: Divendres, 23 de febrer de 2018
Lloc: Aula 005, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Florentino Borondo, Universidad Autónoma de Madrid
Títol: An example of chaotic scattering with 3dof: Atom-surface scattering at nonzero temperature
Resum: In this seminar, we study the chaotic scattering of a He atom off a Cu surface at nonzero temperature using a 3dof Hamiltonian model. The model is a natural extension of a 2dof model that includes the effect of the temperature as a harmonic vibration of the Cu surface. To study the 3dof model it is convenient to analyse first the uncoupled 2dof model for different values of the incident initial energy. We calculate the set of singularities of the scattering functions and study its connection with the tangle between the stable and unstable manifolds of the central fixed point in the Poincaré map for different values of the initial energy. Knowing the stable and unstable manifolds of the 2dof model it is possible to construct the stable and unstable manifolds for the model that includes the vibration of the surface with 3dof. It is a deformation of a stack of the 2dof system. Also, for the 3dof system the resulting invariant manifolds have the correct dimension to divide the phase space. By this construction we can understand the scattering phenomena in analogy to those in 2dof Hamiltonian systems. In particular, we explain the connection between the set of singularities of the scattering function and the cross section and their behaviour when the coupling with the vibration is switched on
Dia: Dimecres, 14 de març de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Josep Sardanyès, Centre de Recerca Matemàtica (CRM)
Títol: Some insights into stochastic drift on Normally Hyperbolic Invariant Manifolds
Resum: Recent theoretical research on the within-cell dynamics of viral RNAs under differential replication modes has revealed the presence of so-called quasi-neutral scenarios which are governed by Normally Hyperbolic Invariant Manifolds (NHIMs). Under determinism, these invariant manifolds involve that different asymptotic states are achieved depending on the initial conditions. Under stochasticity, however, these dynamics are broken and the stochastic trajectories travel along this manifold until a given absorbing or asymptotic state is achieved. In this seminar we will discuss recent theoretical and computational results on noise-induced drift in one-dimensional NHIMs, identified in models of competing species as well as in disease dynamics.
We will discuss results on this diffusive theory applied to the model for viral RNA dynamics, which are in agreement with stochastic Gillespie simulations. A novel mechanism for noise-induced bistability will be introduced. Finally, we will discuss some ongoing research aiming at developing a diffusive theory for two-dimensional NHIMs.
Dia: Dimecres, 21 de març de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Otávio Gomid, Universidade Estadual de Campinas i UPC
Títol: Critical Energy in Soliton-Defect Interaction Models
Resum: In this talk, we present a toy-model for interactions between kinks (solitons) of the sine-Gordon equation and a weak defect (a small perturbation modeled by a Dirac delta function). We consider a finite-dimensional reduction of the sine-Gordon equation which is given by a 2-degrees of freedom Hamiltonian H, and we propose a geometric approach to give conditions on the energy of the system to admit kinks. More precisely, we obtain an asymptotic expression of the critical energy hc for which the system admits kinks with small amplitude for energies h≥hc. Our methods rely on computing the exponentially small transversality of invariant manifolds Wu,s of certain objects (critical points and periodic orbits) at infinity.
This is a joint work with M. Guardia and T. M. Seara.
Dia: Dimecres, 4 d'abril de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Mercè Ollé, Universitat Politècnica de Catalunya
Títol: The hydrogen atom in a circularly polarized microwave field: Hopf bifurcation and chaos
Resum: We consider the CP problem, i.e., the hydrogen atom in a rotating electric field, whose dynamics is described by a Hamiltonian of two degrees of freedom depending on one parameter, K>0. We analyse the Hopf bifurcation appearing around one of the equilibrium points when K crosses a critical value Kcrit. The effect of this bifurcation, focusing on regular and bounded motion versus chaotic one is also discussed. This is a joint work with Juan R. Pacha
Dia: Dimecres, 11 d'abril de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Carles Simó, Universitat de Barcelona
Títol: A perturbed Hamilton-Hopf model: splitting, dominant harmonics, splitting volume and hidden harmonics
Resum: We consider a Hamiltonian with an equilibrium having a Hopf bifurcation for ν=0 and becoming a complex saddle for ν>0. After normalization and scaling, with coefficients giving rise to compact invariant manifolds, we add a very simple perturbation depending on position and periodic in time, but which contains infinitely many terms.
We study the splitting after the perturbation, mainly using first order Poincaré-Melnikov approach. After displaying some information on the nodal lines and its bifurcations we look for the changes in the relevant harmonics in the splitting of the manifolds, on the splitting volume at homoclinic points and on the harmonics associated to best approximants which never dominate the splitting functions (hidden harmonics).
We study different cases for the frequency of the time periodic perturbation. This leads, in a natural way, to several open questions in number theory.
This is part of a joint work with Ernest Fontich and Arturo Vieiro.
Dia: Dimecres, 18 d'abril de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Alexandre Rodrigues, University of Porto
Títol: Strange attractors near a homoclinic cycle to a bifocus
Resum: In this seminar, we explore the chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions.
We extend previous results on the field and we show that, when the cycle is broken, there are parameters for which the first return map to a given cross section exhibits homoclinic tangencies associated to a dissipative saddle periodic point. These tangencies can be slightly modified in order to satisfy the Tatjer conditions for a generalized tangency of codimension two. This configuration may be seen the organizing center, by which one can obtain strange attractors and infinitely many sinks.
Therefore, the existence of a homoclinic cycle associated to a bifocus may be considered as a criterion for four-dimensional flows to be 𝒞 ¹-approximated by other flows exhibiting strange attractors
Dia: Dimecres, 2 de maig de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Arturo Vieiro, Universitat de Barcelona
Títol: Dynamics of near identity maps and interpolating vector fields
Resum: We shall introduce the interpolating vector fields as a tool to explore the dynamics of near identity maps. These vector fields are constructed directly from the iterates of the map. Their properties and their relation with averaging will be discused. Several examples will be given. In particular, we shall illustrate how they can be used in a quantitative study of a 4-dimensional map near a double resonance where we use an interpolating vector field to construct an adiabatic invariant for the map. Moreover, they provide a natural way to compute iterates of "Poincaré maps" in the discrete setting, allowing to obtain useful 2-dimensional and 3-dimensional visualizations of the 4-dimensional dynamics.
This is a joint work with Vassili Gelfreich (Univ. Warwick, UK).
Dia: Dimecres, 9 de maig de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Santiago Ibáñez, Universidad de Oviedo
Títol: Homoclinic structures in the Hindmarsh-Rose neuron model
Resum: Outstanding problems in neuroscience involve many different scientific fields including, of course, Dynamical Systems. Among these challenges, the understanding of single neurons is maybe one of the basics. From the dynamical systems point of view, many of the models which are used to describe the functioning of a single neuron are not yet completely understood. The Hindmarsh-Rose is one of those models, particularly popular because, among a great variety of behaviours, it includes the bursting phenomena: a common and essential behaviour in biological neurons. HR model is a fast-slow system, with two fast and one slow variables. Izhikevich (see [1]) provided a complete description of all possible interplays between bifurcations which can explain the emergence of bursting in 2+1 fast-slow systems. Following his classification, HR model displays two cases: fold/homoclinic and fold/Hopf bursting. However, the overall structure of the involved bifurcation diagram has not been fully understood yet. For that reason, focusing on the fold/homoclinic bursting, we provide (see [2,3]) a global analysis of the homoclinic bifurcations structure. We consider all parameters fixed but three and, after a deep numerical analysis in the 3-parameter space, we are able to conjecture a theoretical organization of the whole structure.
In this talk we will describe the theoretical model and the numerical results that support it. It includes surfaces of codimension 1 homoclinic bifurcations, curves of codimension 2 homoclinic bifurcations (Belyakov, inclination and orbit flips) and also higher codimension homoclinic bifurcations. Foldings of bifurcation surfaces and bifurcation curves in the parameter space also explains the appearance of isolas of homoclinic bifurcation.
This is a joint work with R. Barrio and L. Pérez.
Dia: Dimecres, 16 de maig de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Ariadna Farrés, NASA, Goddard Space Flight Center, Navigation and Mission Design Branch.
Títol: Solar Radiation Pressure Modeling and its effects at Libration Point Orbits
Resum: Solar Radiation Pressure (SRP) is the acceleration produced by the impact of the Sun light photons on the surface of a satellite. The incident photons are absorbed and reflected by the different components on its surface, where the rate of absorption and reflection depends on the properties of the surface material. The acceleration produced by SRP plays an important role on Libration Point orbits and interplanetary trajectories.
The first part of this talk we will describe the different approaches that appear in the literature to approximate this effect, and we will introduce an alternative way to obtain high fidelity models for the SRP acceleration using a Spherical Harmonic approximation.
The second part of the talk we will discuss the relevance of SRP in the Wide-Field Infrared Survey Telescope (WFIRST). WFIRST is a NASA observatory designed to answer questions about dark energy and astrophysics, planned for a launch in 2025 to orbit around the Sun-Earth L2 (SEL2) Libration Point. One of the instruments on WFIRST is a Coronagraph, whose primary objective is to search for exoplanets. The use of an external occulter such as Starshade would make the detection of Earth-sized planets in habitable zones of nearby stars possible. As we will see SRP plays an important role in the coupled motion of WFIRST and Starshade.
This work has been done in collaboration with Dave Folta and Cassandra Webster from NASA, Goddard Space Flight Center, Navigation and Mission Design Branch.
Dia: Dimecres, 23 de maig de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Jonathan Jaquette, Rutgers University, New Jersey
Títol: Counting and discounting slowly oscillating periodic solutions: a proof of the Jones' conjecture
Resum: An often studied example of a nonlinear delay differential equation is Wright's equation: y'(t)=-αy(t -1)[1+y(t)]. At α= π/2 there is a supercritical Hopf bifurcation, and the Wright/Jones conjecture asserts that there are no slowly oscillating periodic solutions (SOPS) for α<π/2, and there exists a single, unique SOPS for each α>π/2.
While the existence of SOPS for α>π/2 was shown in 1962, proving uniqueness has been more difficult to obtain. This talk presents a computer-assisted proof of these conjectures, focusing in particular on the mesoscopic parameter regime α∈(π/2,1.9]. In our approach, we use a priori estimates to obtain global bounds on the Fourier coefficients, and then we use a "branch and bound" algorithm to ensure that each parameter value has a unique slowly oscillating periodic orbit. Furthermore, we show there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.
Dia: Dimecres, 30 de maig de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Jordi-Lluís Figueras, Uppsala University
Títol: Loss of uniform hyperbolicity and regularity of the Lyapunov exponent
Resum: What happens when a system undergoes a bifurcation from uniform, to non-uniform, hyperbolic behaviour? We consider a certain class of smooth, Diophantine, quasi-periodically forced Schrödinger cocycles, where we can study how the stable and unstable directions break apart under the dynamics. This information can then be used to obtain regularity results for the Lyapunov exponent.
The parameter E we consider is related to the spectrum of the corresponding Schrödinger operator. Almost optimal regularity results, for the Lyapunov exponent L(E) depending on this parameter, are already known for certain classes of analytic Schrödinger cocycles, but in the smooth case the results are fewer and typically not as sharp. For certain parameter values (the lowest energy of the spectrum) we obtain good estimates on the regularity of the Lyapunov exponent, as long the cocycle satisfies a non-degeneracy condition (which is usually imposed to guarantee hyperbolicity).
In 2009, Bjerklov & Saprikina proved that the distance between invariant bundles has linear asymptotics, proving part of a conjecture stated in Haro & de la Llave 2006. We finish proving the conjecture, obtaining that the Lyapunov exponent has square root asymptotics.
Dia: Dimecres, 6 de juny de 2018
Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC.
A càrrec de: Alejandro Luque, Uppsala University
Títol: Stationary phase methods and the splitting of separatrices
Resum: In this talk we discuss the splitting of separatrices under rapidly oscillating (in time and space) perturbations. Using stationary phase methods, we provide an explicit formula for the asymptotic displacement of the invariant manifolds, which becomes algebraically small. Remarkably, the proof does not require any explicit knowledge of the separatrix and works also for non-analytic systems. As a corollary, we obtain positive topological entropy under generic conditions. The perturbations considered are related to a priori stable systems with low regularity perturbations. They model also wave-type perturbations that arise in the study of the motion of charged particles in a rapidly oscillating electromagnetic field.
Dia: Dimecres, 13 de juny de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Anna Gierzkiewicz, Agriculture University in Kraków
Títol: Chaos in Hyperion's rotation: a computer-assisted proof
Resum: The inner rotation of Saturn's moon Hyperion is often modeled by equations of an ellipsoidal satellite on a Keplerian orbit, with rotation axis perpendicular to its plane. The angle of rotation θ fulfills a three-dimensional ordinary differential equation
The model is expected to be chaotic for large range of parameters $e$, ω. The aim of the talk is to present a rigorous computer-assisted proof of chaos in its dynamics by the use of CAPD C++ library. The proof follows the general method from [G. Arioli and P. Zgliczy'ski. Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level. J. Diff. Eq., 171(1):173-202, 2001.
We study the Poincaré map P on the 2-dim section {f=0}. In short, we search for the hyperbolic stationary points of P and transversal intersections of their stable and unstable manifolds. Then we determine some special sets (h-sets) with a loop of covering relations between them. The existence of such loop connecting two hyperbolic points proves rigorously the existence of a Smale horseshoe type set for P and, consequently, the chaotic behaviour of the system.
Dia: Dimecres, 27 de juny de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Daniel Coronel, Universidad Andrés Bello, Santiago, Chile
Títol: Non computable Mandelbrot-like set for a one-parameter complex family
Resum: In this talk we will review some basic concepts in computable analysis and complex dynamics. Next, we will recall what is known about the Mandelbrot set and Julia sets of the quadratic family from the computable analysis point of view. Finally, we show the existence of computable complex numbers for which the bifurcation locus of the one parameter complex family fb(z) = λz + b z² + z³ is not Turing computable.
Joint work with Cristóbal Rojas and Michael Yampolsky.
Dia: Dimecres, 4 de juliol de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Vadim Kaloshin, University of Maryland at College Park
Títol: Can you hear the shape of a drum and deformational spectral rigidity of planar domain?
Resum: M. Kac popularized the question "Can you hear the shape of a drum?". Mathematically, consider a bounded planar domain and the associated Dirichlet problem The set of 's such that this equation has a solution, is called the Laplace spectrum of. Does Laplace spectrum determines? In general, the answer is negative.
Consider the billiard problem inside. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that a generic axis symmetric planar domain with sufficiently smooth boundary is dynamically spectrally rigid, i.e. can't be deformed without changing the length spectrum. This partially answers a question of P. Sarnak.
This is based on two joint works with J. De Simoi, Q. Wei and J. De Simoi, A. Figalli.
Dia: Dimecres, 18 de juliol de 2018
Lloc: Aula T2 (2n pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Piotr Zgliczynski, Jagiellonian University
Títol: A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line
Resum: We discuss a method for rigorous study of dynamics of dissipative PDEs. The method is then applied to certain Poincaré map of the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter ν=0.1212. We give a computer-assisted proof of the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.
This is a joint work with Daniel Wilczak
Last updated: Saturday, 13-Oct-2018 13:01:59 CEST