Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Yannick Sire, INSA Toulouse .
Títol: Weak coupling limits and localized oscillations in some Euclidean lattices .
Lloc: Aula B6 (planta baixa), Facultat de Matemàtiques, UB.
A càrrec de: Tomasz Kapela, Institute of Computer Science, Jagiellonian University, Krakow .
Títol: Rigorous algorithm for integration of differential inclusions .
Resum: We describe a modification of the Lohner algorithm which allows us to compute rigorously bounds for reachable set for control systems, solutions of ordinary differential inclusions and perturbations of ODEs.
A càrrec de: Daniel Wilczak, Institute of Computer Science, Jagiellonian University, Krakow .
Títol: Rigorous numerics for period doubling bifurcations with an application to the Rossler system .
Resum: We introduce a method for rigorous verification that a map admits a period doubling bifurcation. The method has been applied to the Rossler system
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de:Pablo Cincotta, Facultad de Ciencias Astronomicas y Geofisicas, Universidad Nacional de La Plata .
Títol: Testing the accuracy of the overlap criterion .
Resum: Here we investigate the accuracy of the overlap criterion when applied to a simple near-integrable 2D and 3D model. To this end, we consider respectively two and three quartic oscillators as the unperturbed system, and couple the degrees of freedom by a cubic, non-integrable perturbation. For this system we compute the unperturbed resonances up to order O(ε2).
We model each resonance by means of the pendulum approximation and estimate the theoretical critical value for the perturbation parameter for a global transition to chaos. We then perform, for the 2D model, several surface of sections in order to compare our theoretical estimation for such critical value with the resulting empirical one. We use previous numerical estimation of the critical value to compare with the one derived here.
A càrrec de: Claudia Giordano, Facultad de Ciencias Astronomicas y Geofisicas, Universidad Nacional de La Plata .
Títol: Diffusion in phase space of multidimensional Hamiltonian systems .
Resum: In this work we review as well as provide new results on the processes that lead to chaotic diffusion in phase space of multidimensional Hamiltonian systems.
It is well known that the simplest mechanisms leading to a transition from regularity to chaos, and therefore to diffusion in phase space, are the overlap of resonances, resonance crossings, and Arnol'd diffusion-like processes.
When dealing with nearly integrable Hamiltonian systems, chaos actually means the variation of the unperturbed integrals, which is usually called chaotic diffusion. Unfortunately, it does not yet exist any theory that could describe global diffusion in phase space. In other words, it is not possible to estimate either its routes or its extent. Though one could get accurate values of the Lyapunov exponents, the KS entropy, or any other indicator of the stability of the motion, they only provide local values for the variation of the integrals. A given orbit in a chaotic component of phase space could have, for instance, a positive and large value for two of the Lyapunov exponents, however, this does not necessarily mean that the unperturbed integrals would change over a rather large domain. This is a natural consequence of the structure of phase space of almost all actual dynamical systems such as planetary systems or galaxies.
Therefore, what is actually significant is the extent of the domain and the time-scale over which diffusion may occur. In a previos work it is shown that in models similar to those suitable for the description of an elliptical galaxy, the time-scale over which diffusion becomes relevant is several orders of magnitude the Hubble time. On the other hand, in models corresponding to planetary or asteroidal dynamics, diffusion may occur in physical time-scales.
All these issues are thoroughly discussed here by both numerical and theoretical means.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Luis Benet, UNAM .
Títol: Migration and eccentricity in simple planetary accretion models .
Resum: Recent observations of exosolar systems have shown the surprising existence of Jupiter-like exoplanets in orbits which are too close to their host star. One possibility to explain this is planetary migration.
In this talk, using a simple statistical model for planetary accretion we address the question on migration of an early formed hot Jupiter within a disk of planetesimals. We distinguish two scenarios depending on the time-scales involved. The first one, where accretion takes place everywhere in the disk with the same probability. The second one, where two distinct time scales define the accretion process: a short time scale related to accretion with the massive planet, and a slow one related with accretion elsewhere. Both scenarios lead to migration; yet, the final eccentricity of the massive planet displays distinct statistical results. The relation of our results with the experimental observations will be addressed.
A càrrec de: Eric Lombardi, CNRS, Laboratoire MIP, Univ. Paul Sabatier, Toulouse .
Títol: Homoclinic connections, complex singularities and astronomers's summation .
Resum: We are interested in the existence of homoclinic connections to 0, for reversible vector fields V(u,μ) in R4 admitting a 02iω resonance at the origin i.e. vector field for which for μ=0 the linear part at the origin admits 0 as double eigenvalue and ±iω as simple eigenvalues. The normal forms at any order of such vector fields do admit homoclinic connections to 0. However, I will present two differents methods to prove that in fact generically there is no homoclinic connections to 0 but always homoclinic connections to exponentially small periodic orbits. The first method is based on the study of the complex continuation of the solutions, and the second on an optimization of the order of the normal form to get an exponentially small remainder
Lloc: Aula IMUB (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Henk A. Dijkstra, Institute for Marine and Atmospheric Research, Dept. of Physics and Astronomy, U. Utrecht .
Títol: A tailored solver for bifurcation analysis of ocean GCMs .
Resum: To determine the bifurcation behavior of solutions of either global ocean-climate models or high-resolution regional ocean models, one must be able to follow branches of steady states versus parameters in very large dimensional dynamical systems (more than 1 million degrees of freedom). The central problems are
Recently, we have made major steps forward on these problems by developing a tailored ocean model solver. This solver takes special advantage of the mathematical structure of the governing equations and uses a hybrid (partly analytical/numerical) evaluation of the Jacobian matrix. With this solver we can now determine fixed points (and their linear stability) of quite sophisticated ocean models. In this talk, I will give an overview of the ideas behind the tailored solver and illustrate its capabilities.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Arsen Dzhanoev, Physics Faculty, Lomonosov Moscow State University i Departamento de Fisica, Universidad Rey Juan Carlos .
Títol: Stabilized orbits in the restricted three-body problem .
Resum: A new type of orbit in the restricted three-body problem is constructed. It is analytically shown that along with the well known chaotic and regular orbits in the three-body problem there also exists a qualitatively different type of orbit which we call "stabilized". The stabilized orbits are a result of additional orbiting bodies that are placed in the triangular Lagrange points. The results are well confirmed by numerical orbit calculations.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Rick Moeckel, Math. Dept. Univ. of Minnesota and IMCCE .
Títol: A Topological Existence Proof for a Figure-Eight Orbit of the Three-Body Problem .
Resum: I will describe a topological existence proof for a figure-eight periodic solution of the equal mass three-body problem. The proof is based on the construction of an Wazewski set W in the phase space. The figure-eight solution is then found by a kind of shooting argument in which symmetrical initial conditions entering W are followed under the flow until they exit W. A linking argument shows that the image of the symmetrical entrance states under this flow map must intersect an appropriate set of symmetrical exit states.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Luca Zampogni, Dipartimento di Matematica e Informatica, Universita di Perugia .
Títol: On a Class of Reflectionless Sturm-Liouville Potentials .
Resum: The Sturm-Liouville equation −(pφ')'+qφ=λyφ has been studied for about two centuries. When p=y=1, we have the well-known Schrödinger equation, which received a great attention. One often uses the adjectives “algebro-geometric, reflectionless, Sato-Segal- Wilson (SSW)” (and others as well) to describe particular families of Schrödinger potentials q. Moreover, Schrödinger potentials are linked to the solutions of the Korteweg-de Vries equation
Particular attention will be given to an inverse spectral problem for algebro-geometric Sturm-Liouville coefficients: namely, given a set Σ which is the union of a finite number of closed intervals and a half-line [r,∞), we will construct stationary ergodic processes (A{τt},μ) of algebro-geometric Sturm-Liouville potentials a=(p,q,y) having Σ as spectrum and vanishing upper Lyapunov exponent in Σ.
We will use algebro-geometric Sturm-Liouville potentials to build large families of reflectionless and SSW Sturm-Liouville potentials and to put in evidence some fundamental properties of the corresponding spectral problems.
Moreover, we will relate the algebro-geometric Sturm-Liouville coefficients to the solutions of the Camassa-Holm (CH) equation
This work is a report on a joint work with Russell Johnson (University of Florence).
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramirez Ros, Dept. Matematica Aplicada I, UPC .
Títol: Escision de separatrices en aplicaciones preservando volumen .
Resum: Empezare construyendo una familia de aplicaciones integrables que preservan volumen con una conexion bidimensional entre dos puntos fijos hiperbolicos. Quiero estudiar diversos aspectos relacionados con la escision de las separatrices bajo perturbaciones preservando volumen. La herramienta basica es una version discreta del metodo de Melnikov.
En primer lugar, describire la forma (y las bifurcaciones) de las intersecciones heteroclinicas primarias para una perturbacion concreta. Despues, presentare varias propiedades mas generales. Por ejemplo, acotare la complejidad topologica de las intesecciones heteroclinicas en funcion del grado de algunas perturbaciones polinomiales. Tambien dare una condicion suficiente para la escision de la separatriz bajo algunas perturbaciones enteras y comprobare que un amplio abanico de perturbaciones polinomiales cumplen dicha condicion.
Creo que el estudio es divertido, no solo por el QUE, sino tambien por el COMO se prueban los resultados: vision geometrica, trucos algebraicos, teoremas de variable compleja, aparicion de funciones cuasi-elipticas, uso de clases de homologia, calculos en multiple precision, etc. Pero todo a nivel ELEMENTAL. Un alumno de carrera de ultimo año podria seguir todos los razonamientos.
Tambien se abren varios frentes interesantes: fenomenos exponencialmente pequeños, transporte, resonancias, toros invariantes, etc.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jean-Philippe Lessard, Dept. of Mathematics, Rutgers University .
Títol: Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation .
Resum: In this talk, we prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semi-conjugacy to a subshift of finite type shows that the dynamics is chaotic. This is joint work with Jan Bouwe van den Berg (VU University Amsterdam).
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Alberto Campos, Departamento de Matemáticas, Universidad Nacional, Bogotá .
Títol: Estudio de ecuaciones diferenciales de cuarto orden mediante grupos de Lie .
Resum: En esta charla estudiaremos propiedades relevantes de ecuaciones diferenciales de cuarto orden obtenidas mediante el metodo de Lie.
A càrrec de: David Farrelly, Utah State University i UAM .
Títol: Chaotic capture of Kuiper Belt binaries .
Resum: The discovery that many trans-Neptunian objects are binaries is invaluable for shedding light on the formation, evolution and structure of the outer Solar system, e.g., the dynamics of debris disks. More than 20 Kuiper-belt binaries (KBBs) are now known. These relics from the primordial Solar System differ in several key ways from other known populations of binary objects, e.g., most KBBs consist of similarly-sized partners which are following large, eccentric mutual orbits. It is proposed that chaos played a significant role in the formation of KBBs and that the fingerprint of chaos is visible in the orbital and physical properties of these objects.
Our calculations suggest that binaries are produced through the following chain of events. Initially, long-lived quasi-bound binaries form when two bodies get caught-up in thin layers of dynamical chaos produced by solar tides. Gravitational scattering then stabilizes and hardens the binary. Predictions of the model and comparison with recent observations will be made.
Extensions of this mechanism to other problems of interest in the outer Solar System and to problems in atomic and molecular physics will be discussed.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Vassili Gelfreich, Math. Dept., Warwick University .
Títol: Slow drift in a slow-fast Hamiltonian system .
Resum: We consider a slow-fast Hamiltonian system with several fast and slow degrees of freedom. Assuming the fast system with the frozen slow variables has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories, we prove that for any path composed of a finite sequence of slow trajectories generated by action Hamiltonians, there is a trajectory of the full system whose slow component shadows the path.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Jacques Fejoz, Univ. Paris 6 i Observatoire de Paris .
Títol: Unchained polygons and the n-body problem .
Resum: Look to a relative equilibrium of the n-body problem in a rotating frame which puts into resonance the frequency of a relative equilibrium of the n-body problem and that of an infinitesimal variation normal to the plane of the equilibrium. Continuation with respect to the rotating velocity of the frame yields a remarkable class of periodic solutions. The first example is the P12 family, discovered by C. Marchal, which links the relative equilibrium of Lagrange to the Eight. Other examples are given by the continuation of the n-gon.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Anatoly Neishtadt, Loughborough University i Space Research Institute (Moscow) .
Títol: Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems .
Resum: We consider a 2 d.o.f. Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of the time derivatives ofthe slow and fast variables is of order ε << 1. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ε) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ε . So, the total measure of the stability islands is estimated from below by a value independent of ε. We find an asymptotic formula for the number of stable periodic trajectories. As an example, we consider the problem of motion of a charged particle in the parabolic model of magnetic field in the Earth magnetotail. This is a work in collaboration with C. Simó, D. Treschev and A. Vasiliev.
A càrrec de: Lluís Alseda, Univ. Autònoma de Barcelona .
Títol: Attractors for unimodal quasiperiodically forced maps .
Resum: We consider pinched unimodal quasiperiodically forced maps, that is, skew products with irrational rotations of the circle in the base and unimodal interval maps in the fibers. This case is similar to the one considered by Gerhard Keller, except that, in his case, the function in the fibers is increasing.
We prove that under some additional assumptions on the system there exists a "strange nonchaotic attractor". It is the graph of a measurable function from the circle to a closed interval of the real line which is invariant, discontinuous almost everywhere and attracts almost all trajectories. Moreover, both Lyapunov exponents on this attractor are nonpositive. There are also cases when the dynamics is completely different, because one can apply the results of Jerome Buzzi implying the existence of an invariant measure absolutely continuous with respect to the Lebesgue measure (and then the attractor is some region in the cylinder), and the maximal Lyapunov exponent is positive. Finally, there are cases in which we can only guess what the behavior is by performing computer experiments. This is a work in collaboration with M. Misurewicz.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Alan R. Champneys, Dept of Engineering Mathematics, University of Bristol .
Títol: Localisation and beyond-all-orders effects .
Resum: This talk shall examine a range of problems where nonlinear waves or coherent structures are localised to some portion of a domain. In one spatial dimension, the problem reduces to finding homoclinic connections to equilibria. Two canonical problems emerge when higher-order spatial terms are considered (either via fourth-order operators or discreteness effects). One involves so-called snaking bifurcation diagrams where a fundamental state grows an internal patterned layer via an infinite sequence of fold bifurcations. The other involves the exact vanishing of oscillatory tails as a parameter is varied. It is shown how both problems arise from certain codimension-two limits where they can be captured by beyond-all- orders analysis. Dynamical systems methods can then be used to explain the kind of structures that emerge away from these degenerate points. Applications include moving discrete breathers in atomic lattices, discrete solitons in optical cavities, and theories for two-dimensional localised patterns using Swift-Hohenberg theory.
A càrrec de: Sergey Gonchenko, Research Institute for Applied Math. and Cybernetics, Nizhny Novgorod .
Títol: On global bifurcations leading to Lorenz-like wild attractors .
Resum: We consider three-dimensional diffeomorphisms having either homoclinic tangencies or non-transversal heteroclinic cycles. We select special cases of such systems whose bifurcations lead to the appearance of strange wild hyperbolic attractors: here, strange attractors allowing homoclinic tangencies but such that any close system has no stable periodic orbits. Our attractors are Lorenz-like ones since they can be obtained as small periodic perturbations of the Lorenz attractors for flows.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Marco Antonio Teixeira, UNICAMP, Brasil .
Títol: Families of Minimal sets in Reversible Systems .
Resum: We study the dynamics near an equilibrium p0 of a Z2 reversible vector field in R2n with reversing symmetry φ satisfying φ2 = Id and dim Fix(φ) = n. One of the characteristics properties of reversible systems is that generically periodic orbits or invariant tori or minimal sets of such systems typically appear in one-parameter families.
Questions can be formulated, such as:
Our main concern is to find conditions for the existence of one-parameter families of minimal sets (periodic orbits and homoclinic orbits) going through an equilibrium point.
A càrrec de: Alejandro Luque, Dept. Matematica Aplicada I, UPC .
Títol: Cálculo de números de rotación y derivadas respecto a parámetros .
Resum: Recientemente se ha desarrollado un método numérico para aproximar números de rotación de difeomorfismos del círculo con mucha precisión. Dicho método se sirve de la existencia de una conjugación (analítica) a una rotación rigida y, básicamente, consiste en promediar adecuadamente los iterados de la aplicación y extrapolar.
En la primera parte de la charla revisaremos rápidamente dicho método y presentaremos una extensión para calcular derivadas del número de rotación en familias de aplicaciones del círculo. Se mostrarán algunos resultados numéricos relativos a la bien conocida familia de Arnold.
En la segunda parte de la charla queremos proponer un método numérico para construir un difeomorfismo del círculo asociado a una curva invariante de una aplicación general en el plano. Trabajo conjunto con Jordi Villanueva.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Ernest Fontich, Dept. Matemàtica Aplicada i Anàlisi, Univ. Barcelona .
Títol: The parameterization method for invariant tori .
Resum: We present some results which provide the existence of (not necessarily maximal) invariant tori as the image of an embedding from the standard torus to the phase space. The hypotheses are that we have a good approximation of the torus satisfying some non-degeneracy conditions. We do not assume that the system is in action-variable coordinates nor that it is close to integrable. The results extend to some systems defined on lattices provided we have some decay properties for the interaction between nodes.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Núria Fagella, Dept. Matemàtica Aplicada i Anàlisi, Univ. Barcelona .
Títol: Cirurgia en sistemes dinàmics complexos (sense sang) .
Resum: Després de les primeres aplicacions de la cirurgia quasiconforme a començaments dels 80, Douady i Branner al 87 van portar la tècnica més enllà i van proposar una cirurgia que relacionava polinomis de graus diferents, definint així una aplicació entre diferents espais de paràmetres. Per veure que tenien un homeomorfisme, van haver de preocupar-se de questions delicades com la continuïtat d'aquests procediments quirúrgics respecte a paràmetres. En aquesta xerrada esboçarem aquestes tàcniques i veurem com s'han aplicat posteriorment per obtenir altres resultats interessants.
Lloc: Aula T2 (2on pis), Facultat de Matemàtiques, UB.
A càrrec de: Alexei Tsygvintsev, École Normale Supérieure de Lyon .
Títol: The Rattleback and its integrability .
Resum: The rattleback's amazing mechanical behaviour can be described as follows: when spun on a flat horizontal surface in clockwise direction this top continues to spin in the same direction until it consumes the initial spin energy; however, when spun incounter clockwise direction, the spinning soon ceases, the body briefly oscillates, and then reverses its spin direction and thus spins in the clockwise direction until the energy is consumed. The first mathematical model of this phenomena belongs to Walker (1896) who studied the linearized equations of motion and concluded that the completely stable motion is possible in only one (say clockwise) spin direction. This classical explanation of the rattleback's behavior is quite unsatisfactory since it does not reflect the global dynamical effects explaining the transfer of trajectories from the vicinity of the unstable solution to the stable one. We mention that actually only numerical evidence for nonintegrability of rattleback systems has been observed. To analyze thoroughly this question we propose to study the non-holonomic equations of the rattleback and particularly to investigate the existence of complex analytic first integrals in the neighbourhood of some particular solutions.
A càrrec de: Xavier Jarque, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Dynamical hairs as external rays for some entire maps .
Resum: We consider the families of entire transcendental maps given by Fλ,m(z)=λzmexp(z) where m≥2. All functions Fλ,mhave a superattracting fixed point at z=0 and a critical point at z=-m. In the parameter planes we focus on the capture zones, i.e., we consider λ values for which the critical point belongs to the basin of attraction of z=0. Using symbolic dynamics we investigate the connection between the dynamics near zero and the dynamics near infinity in the boundary of the immediate basin of attraction of the origin.
This is a joint work with A. Garijo (URV) i M. Moreno-Rocha (CIMAT,Mexico).
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