Dia: Dimecres, 14 de setembre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
També ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Jaume Paradela (UPC)
Títol: Unstable Motions in the Restricted Planar Elliptic Three Body Problem
Resum: Consider the restricted planar elliptic 3 body problem (RPE3BP), where a massless body moves on the gravitational field generated by two bodies, called primaries, which have masses μ∈(0,1/2] and 1-μ and revolve one around each other in Keplerian ellipses of eccentricity ε∈(0,1). We prove that the RPE3BP exhibits unstable dynamics: for any value μ∈ (0,1/2), we build orbits along which the angular momentum G of the massless body experiences an arbitrarily large variation provided the eccentricity of the primaries is small enough. This kind of unstable motion is also referred to as Arnold diffusion in the literature.
The proof relies on the existence of a transition chain of heteroclinic orbits to a Topological Normally Hyperbolic Invariant Cylinder (TNHIC). Two main sets of tools are developed to construct this diffusive mechanism in the RPE3BP. First, for the study of highly anisotropic splitting (involving exponentially and non exponentially small directions) between the invariant manifolds of pairs of partially hyperbolic invariant tori. Second, for the analysis of two distinct scattering maps associated to two different transversal intersections of the invariant manifolds of a (Topological) NHIC. In particular, for comparing them when they are exponentially close to each other. This is a joint work with Marcel Guardia and Tere Seara.
Dia: Dimecres, 28 de setembre de 2022
Lloc: Aula B1 (planta baixa), Facultat de Matemàtiques i Informàtica, UB.
També ONLINE https://ub-edu.zoom.us/j/99202881559?pwd=cDBlY3d4TS9Xam56bXVIU3c1TmtRUT09
A càrrec de: Pablo Roldán (Yeshiva University)
Títol: Continuation of relative equilibria in the n–body problem to spaces of constant curvature
Resum: The curved n-body problem is a natural extension of the planar Newtonian n-body problem to surfaces of non-zero constant curvature. We prove that all non-degenerate relative equilibria of the planar problem can be continued to spaces of constant curvature κ, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. For example, we extend Lagrange’s triangle configuration with different masses to both positive and negative curvature spaces.
Dia: Dimecres, 19 d'octubre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
També ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Alexey Kazakov (HSE University Nizhny Novgorod)
Títol: Four-winged and two-winged Lorenz attractors emerge due to bifurcations of a periodic orbit with multipliers (-1,i,-i)
Resum: We show that bifurcations of periodic orbits with multipliers (-1,i,-i) can lead to the birth of pseudohyperbolic Lorenz-like attractors of two different types: one is a discrete analogue of the classical Lorenz attractor and the other is new, a four-winged Simó angel. We also show that these two attractors exist for an orientation-reversing, quadratic, three-dimensional Hénon map, which implies the abundance of such attractors in a class of systems with homoclinic tangencies. The analysis is based on the study of a normal form for this bifurcation, a slow-fast three-dimensional system of differential equations with a Z4 symmetry. The existence of the continuous-time counterparts of the discrete 2- and 4-winged Lorenz-like attractors is established for the normal form as a part of an extensive numerical and theoretical analysis of its bifurcations. In particular, we establish that both 2- and 4-winged continuous-time Lorenz attractors are born out of a certain Z4-symmetric heteroclinic configurations with 3 saddles.
Dia: Dimecres, 26 d'octubre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
També ONLINE https://ub-edu.zoom.us/j/99202881559?pwd=cDBlY3d4TS9Xam56bXVIU3c1TmtRUT09
A càrrec de: Donato Scarcella, Paris-Dauphine University
Títol: Asymptotically quasiperiodic solutions for time-dependent Hamiltonians
Resum: In 1954 Kolmogorov laid the foundation for the so-called KAM theory. This theory shows the persistence of quasiperiodic solutions in nearly integrable Hamiltonian systems. It is motivated by classical problems in celestial mechanics, such as the n-body problem.
In this talk, we are interested in perturbations which depend on time non-quasiperiodically. We will analyze some properties of time-dependent Hamiltonians converging, when time tends to infinity, to autonomous Hamiltonians having an invariant torus supporting quasiperiodic solutions. We will look for solutions converging, asymptotically in time, to the quasiperiodic solutions of the unperturbed autonomous system.
Moreover, we will analyze the example in celestial mechanics of a planetary system perturbed by a given comet coming from and going back to infinity, asymptotically along a hyperbolic Keplerian orbit.
Dia: Dimecres, 2 de novembre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
També ONLINE https://meet.google.com/zvg-pajn-owr
A càrrec de: Renato Calleja (IIMAS-UNAM, Universitat de Barcelona)
Títol: Quasi-periodic attractors up to the breakdown in the spin-orbit problem
Resum: We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and drift. This problem is an example of a conformally symplectic system, which is characterized by the property to transform the symplectic form into a multiple of itself. We construct and continue quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. The construction is based on a KAM theorem for conformally symplectic systems, which also provides estimates on the breakdown threshold of the invariant attractor. To construct the invariant attractor, we will use high precision numerical simulations to compute some of the required quantities. The algorithms are guaranteed to reach arbitrarily close to the border of existence, given enough computer resources. This talk refers to joint works with A. Celletti, J. Gimeno and R. de la Llave.
Dia: Dimecres, 9 de novembre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Andrew Clarke, Universitat de Barcelona
Títol: Why are inner planets not inclined?
Resum: Consider the 4-body problem with arbitrary masses in the regime where 3 bodies revolve around the other. We assume that the semimajor axes of the orbital ellipses are of different orders, and that there is non-negligible mutual inclination between the orbital planes of bodies 1 and 2. We prove that, given any finite itinerary of the angular momentum vector of body 2, there exist orbits of the 4-body problem shadowing this itinerary with arbitrary precision. From a geometric point of view, this implies that the eccentricity of the orbit of body 2 and the mutual inclination of the orbital planes of bodies 2 and 3 can be made to follow any finite itinerary. For example, the second planet can flip from prograde to retrograde nearly horizontal revolutions and back again arbitrarily many times.
Dia: Dimecres, 16 de novembre de 2022
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Makrina Agaoglou (ICMAT)
Títol: Transport processes and chemical reaction dynamics: The case of a Potential Energy Surface with four wells and an index-2 saddle
Resum: In this talk, we will explore the phase space structures governing isomerization dynamics on a potential energy surface with four wells and an index-2 saddle. For this model, we analyze the influence that coupling both degrees of freedom of the system and breaking the symmetry of the problem have on the geometrical template of phase space structures that characterizes reaction, using the method of Lagrangian Descriptors.
*Esta charla está parcialmente soportada con la Beca Leonardo a Investigadores y Creadores Culturales 2022 de la Fundación BBVA. La Fundación BBVA no se responsabiliza de las opiniones, comentarios y contenidos incluidos en este seminario los cuales son total y absoluta responsabilidad de sus autores.*
Dia: Dimecres, 23 de novembre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Marina Vegué, Universitat Politècnica de Catalunya
Títol: Firing rate and synaptic weight distributions in plastic networks of spiking neurons
Resum: Networks of spiking neurons have been widely used as models to represent neuronal activity in the brain. These models are reasonably realistic but they are also difficult to treat analytically. Mean-field theory has nevertheless proven to be successful as a method for deriving some of their statistical properties at equilibrium such as the distribution of firing rates. However, these models assume a static connectivity, whereas the connection strengths in real brain networks evolve in time according to plasticity rules that depend on the neuronal activity.
In this talk I will present a way to extend the mean-field formalism to networks with synaptic weights that are prone to plastic, activity-dependent modulation. The plasticity in the model is mediated by the introduction of spike traces. A trace associated to one neuron represents a chemical signal that is released every time the neuron emits a spike and which is degraded over time. The temporal evolution of the trace is controlled by its degradation rate (i.e., a measure of how fast the spiking "memory" is lost) and by the neuron's firing rate.
The mean-field formalism provides a set of equations whose solution specifies the firing rate and synaptic weight distributions at equilibrium. The equations are exact in the limit of traces with infinite memory but I will show that they already provide accurate results when the degradation rate lies within the physiological range. Overall, this work offers a way to explore and better understand the way in which plasticity shapes both activity and structure in neuronal networks.
Dia: Dimecres, 14 de desembre de 2022
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Michel Orieux, Universitat Politècnica de Catalunya
Títol: Control theory for selective neuronal communication.
Resum: On many scales in the brain, from single neurons to populations of tens of thousands and even in whole brain models, oscillatory behaviors can be scouted. In recent years, many tools from dynamical systems have been developed in order to study these oscillators. In this talk, I will expose how to take advantage of this work, and advances in optimal control theory to establish selective communications between populations of neurons modeled by a set of differential equations.
Dia: Dimecres, 11 de gener de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Francisco Marques, Universitat Politècnica de Catalunya
Títol: Extensional channel flow: a beautiful example for dynamical systems theory.
Resum: Extensional self-similar flows in a channel are explored numerically for arbitrary stretching-shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space.
Dia: Dimecres, 18 de gener de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Anna Florio (Université Paris Dauphine-PSL)
Títol: Universal dynamics in 3D stationary Euler flows
Resum: Understanding the dynamical complexity of an ideal fluid has motivated many research works. According to Arnold's vision, the dynamics of stationary Euler flows should be as complicated as those in celestial mechanics. In a joint work with Pierre Berger and Daniel Peralta-Salas, we recover Arnold's vision by showing the existence of a locally dense set of stationary solutions of the Euler equations in ℝ³ made up of universal vector fields. For the proof we introduce new perturbative methods in the context of Beltrami fields in order to import tools from bifurcation theory.
Dia: Dimecres, 15 de febrer de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Kostiantyn Drach, Institute of Science and Technology Austria (ISTA).
Títol: Rigidity of rational maps
Resum: In one-dimensional complex dynamics, a branch of dynamical systems that studies iterations of holomorphic maps, the rigidity question has the classical form: Under which conditions can one promote topological conjugacy between a pair of maps to an analytic (conformal) conjugacy? We call this 'parameter rigidity', as it allows us to distinguish maps within parameter space starting with some 'soft' topological (or even combinatorial) data. On the other hand, for a given map, there is a parallel 'dynamical rigidity' question: Can one distinguish individual orbits in combinatorial terms (e.g. via symbolic dynamics)? For polynomials, and especially for quadratic polynomials, this circle of questions is well-studied in the works of Avila, Douady, Hubbard, Lyubich, McMullen, Sullivan, van Strien, Yoccoz, and many others. In this case, the progress was possible thanks to the fact that most polynomials have a naturally defined Markov partition in their dynamical plane (via so-called Yoccoz puzzles). This is not true for general rational maps, and hence life becomes more complicated (or interesting?). In my talk, I will discuss what we know so far about dynamical and parameter rigidity of general rational maps. Our guiding example will be Newton maps, a family of maps naturally arising from Newton's root-finding method. I will also outline a 'toolbox' of techniques useful to attack these types of questions; the key 'tool' is a renormalization concept of complex box mapping. The main message of the talk hopefully will be that life is not that complicated outline a 'toolbox' of techniques useful to attack these types of questions; the key 'tool' is a renormalization concept of complex box mapping. The main message of the talk hopefully will be that life is not that complicated after all (under certain conditions).
A càrrec de: Stefanella Boatto, Dept of Applied mathematics, Institute of Mathematics, Federal University of Rio de Janeiro
Títol: A journey into the mathematical universe: Topological Data Analysis. A brief introduction and some applications.
Resum: Data Science is a highly evolving discipline that presents many challenges. For example, in the case of an epidemic, how can we identify which socio-economic and social parameters, tourist mobility, prevention and control measures have the greatest impact on the dynamics of an epidemic? In recent years Topological Data Analysis has proved to be a powerful tool, a lens to highlight some dominant features of collective phenomena and explore large and complex data sets. In this lecture we will see a brief introduction and how this new discipline is highly multidisciplinary within mathematics itself, bringing together areas such as Space-Time Analysis, Algebraic Topology, Complex Networks, Alg. Linear and Computation.
Dia: Dimecres, 22 de febrer de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Jaime Paradela (UPC)
Títol: Oscillatory motions in the Restricted 3 Body Problem: A functional analytic approach.
Resum: A fundamental question in Celestial Mechanics is to analyze the possible final motions of the Restricted 3-body Problem, that is, to provide a qualitative description of its complete (i.e. defined for all time) orbits as time goes to infinity. According to the classification given by Chazy back in 1922, a remarkable possible behaviour is that of oscillatory motions, where the motion of the massless body is unbounded but returns infinitely often inside some bounded region. In contrast with the other possible final motions in Chazy’s classification, oscillatory motions do not occur in the 2-body Problem, while they do for larger numbers of bodies. A further point of interest is their appearance in connection with the existence of choaotic dynamics. In this paper we introduce new tools to study the existence of oscillatory motions and prove that oscillatory motions exist in a particular configuration known as the Restricted Isosceles 3-body Problem (RI3BP) for almost all values of the angular momentum. Our method, which is global and not limited to nearly integrable settings, extends previous results by Guardia, Paradela, Seara and Vidal by blending variational and geometric techniques with tools from nonlinear analysis such as topological degree theory. To the best of our knowledge, the present work constitutes the first complete analytic proof of existence of oscillatory motions in a non perturbative setting.
This is joint work with Susanna Terracini.
Dia: Dimecres, 1 de març de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Chongchun Zeng, Georgia Institute of Technology
Títol: Capillary Gravity Water Waves Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping
Resum: We consider the 2-dim capillary gravity water wave problem - the free boundary problem of the Euler equation with gravity and surface tension - of finite depth x₂∈ (-h,0) linearized at a uniformly monotonic shear flow U(x₂). Our main focus are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of the velocity and obtain stronger decay for the remainders.
This is a joint work with Xiao Liu.
Dia: Dimecres, 8 de març de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Frank Trujillo (Universität Zürich)
Títol: Ejemplos de inestabilidad para hamiltonianos con dos grados de libertad.
Resum: Es sabido que los puntos fijos elípticos genéricos de hamiltonianos suaves con dos grados de libertad son estables en el sentido de Lyapunov. Más concretamente, son KAM-estables, es decir, acumulados por toros invariantes cuya densidad de Lebesgue tiende a 1 cerca del punto fijo, e isoenergéticos no degenerados, lo que garantiza que cada superficie de energía contiene alguno de los toros invariantes. Se sabe que estas dos condiciones implican la estabilidad del punto fijo.
En esta charla, veremos que la estabilidad KAM por sí sola no es suficiente para garantizar la estabilidad en el sentido de Lyapunov para puntos fijos elípticos de hamiltonianos suaves con dos grados de libertad. Construiremos tales ejemplos para cualquier vector de frecuencia con coordenadas de distinto signo.
Dia: Dimecres, 29 de març de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Robert Cardona (UPM-ICMAT)
Títol: Periodic orbits and surfaces of section on Hamiltonian systems: from contact to stable energy level sets
Resum: The quest for periodic orbits and surfaces of section (or "Birkhoff sections") traces back to the work of Poincaré in his study of the restricted three-body problem. Proving the existence of these two objects on regular energy level sets of Hamiltonian systems has been a leading problem in the flourishing field of symplectic dynamics. In this talk, we are interested in the dynamics of non-vanishing volume-preserving vector fields on closed three-manifolds, equivalently understood as Hamiltonian vector fields along regular energy level sets. We will survey some recent striking results for the class of Reeb vector fields defined by contact forms, and present generalizations to the case of vector fields defined by stable Hamiltonian structures. Such structures appear, for example, when restricting a Hamiltonian vector field to a "stable" energy level set.
This talk is based on joint work with A. Rechtman.
Dia: Dimecres, 12 d'abril de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Santiago Barbieri (Université Paris-Saclay and Università degli Studi Roma Tre)
Títol: Analytic smoothing and Nekhoroshev estimates for Hölder steep Hamiltonians
Resum: In this seminar, I will present the first result of Nekhoroshev stability for steep Hamiltonians in Hölder class. Our new approach combines the classical theory of normal forms in analytic category with an improved smoothing procedure to approximate an Hölder Hamiltonian with an analytic one. It is only for the sake of clarity that we consider the case of Hölder perturbations of an analytic integrable Hamiltonian, but our method is flexible enough to work in many other functional classes, including the Gevrey one. The stability exponents can be taken to be (l-1)/(2n α1 ... αn-2) + 1/2 for the time of stability and 1/(2n α1 ... αn-2) for the radius of stability, n being the dimension, l>n+1 being the regularity and the αi’s being the indices of steepness. Crucial to obtain the exponents above is a new non-standard estimate on the Fourier norm of the smoothed function. As a byproduct we improve the stability exponents in the 𝒞k class, with integer k.
Joint work with J.P. Marco and J. Massetti.
Dia: Dimecres, 19 d'abril de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Gian Marco Canneori (Università di Torino)
Títol: The N-centre problem on Riemannian surfaces: a variational approach.
Resum: The classical N-centre problem of Celestial Mechanics describes the behaviour of a point particle under the attraction of a finite number of motionless bodies. Considered as a limit case of a (N+1)-body problem, it has been the object of several results concerning integrability, investigation of chaos and existence of periodic orbits, mostly when the motion is constrained to the Euclidean plane. In particular, variational approaches are convincing in this situation and have produced classes of collision-less periodic solutions, after imposing topological constraints of different natures. Looking for genuine solutions of second order differential equations, the most delicate step resides in avoiding collisions with the centres. Picturing a more realistic situation, a natural extension of these results could be the one in which the motion is constrained to a prescribed Riemannian surface. In this talk we state the N-centre problem on orientable surfaces and we show how it is possible to use variational arguments in order to produce a huge set of collision-less periodic solutions. Such trajectories will be found among homotopy classes of loops, and their variational and topological properties will be described. Also, the major difficulties and differences with the planar case will be addressed.
This is a joint work with Stefano Baranzini from the University of Torino.
Dia: Dimecres, 26 d'abril de 2023
Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.
A càrrec de: Marcel Guàrdia (Universitat de Barcelona)
Títol: Transfer of energy in Hamiltonian systems on infinite lattices
Resum: In this talk we present a recent result about the existence of transfer of energy orbits in a chain of infinitely many weakly coupled pendulums. The model system is posed on an infinite lattice (of any dimension) with formal or convergent Hamiltonian. We develop geometric tools to perform an Arnold diffusion mechanism in infinite dimensional phase spaces. In this way we construct solutions that move the energy of the pendulums along any prescribed path in the lattice.
This is a joint work with Filippo Giuliani.
Dia: Dimecres, 21 de juny de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Martin Leguil (Université de Picardie Jules Verne)
Títol: Conformally Symplectic Billiards
Resum: In a joint work with Anna Florio and Olga Bernardi, we study the dynamics of conformally symplectic convex billiards, namely, the usual elastic reflection law is replaced with a new law where the angle bends towards the normal after each collision. The dynamics of such billiards exhibits a global attractor; in our work, we focus on an invariant subset of this attractor, the so-called Birkhoff attractor, which was first introduced by Birkhoff in a general framework and whose study has been pursued in several works, in particular by Charpentier and Le Calvez. Our goal is to understand how complex the Birkhoff attractor may be for conformally symplectic convex billiards. We start with the example of conformally symplectic elliptic tables, for which we show that the dynamics is of Morse-Smale type. Moreover, for generic convex billiards, in presence of a mild dissipation, we show that the attractor can be much more complicated, both topologically and dynamically.
A càrrec de: Juan J. Morales-Ruiz (Universidad Politécnica de Madrid)
Títol: Semiclassical quantification of some two degree of freedom potentials: a Differential Galois approach
Resum: This seminar will be devoted to explain the relevance of the differential Galois theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a particular solution depends on the variational equation around that solution: a very well-known object in dynamical systems, general relativity and variational calculus. Then, as the variational equation is a linear ordinary differential system, it is possible to apply the differential Galois theory to study its solvability in closed form. We obtain closed form solutions for the semiclassical quantum fluctuations around free particle type solutions for some systems, like the classical Verhulst’s potentials. We remark that the systems studied are not integrable. We will try to give the essential ideas behind the results, leaving aside the technical details.
This is a joint work with Primitivo Acosta-Húmanez, José Tomás Lázaro and Chara Pantazi.
* Seminars partially supported by a 2022 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The BBVA Foundation accepts no responsibility for the opinions, statements and contents included in the conference and/or the results thereof, which are entirely the responsibility of the authors. *
Dia: Dijous, 28 de juny de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Guillermo Olicón Méndez (Universität Berlin, Germany)
Títol: A random dynamical system perspective on chemical reaction networks
Resum: Given a system of reacting chemicals, the evolution of the number of particles of each of the species involved can be modeled with different approaches, depending on a right scaling with respect to the volume where the reactions take place. On the one hand, when the number of individuals is to be analysed, typically because some of the chemicals are in low concentration, one can adopt the Markov jump process approach. Roughly speaking, these type of stochastic processes remain constant until one of the reactions occur at a random time. On the other hand, when all chemical species scale appropriately with the volume, one may consider the so-called chemical Langevin equations – a set of stochastic differential equations of Itô type.
In this talk we introduce the concept of random dynamical systems (RDS) in the context of chemical reaction networks. Similarly as in deterministic dynamical systems, the RDS perspective aims to study what happens to neighbouring initial conditions when assuming that the noise realisation is fixed. We briefly sketch how both approaches – the jump process and the SDE– induce RDS. Under the presence of noise, the systems may exhibit phenomena which were no presence in their deterministic counterparts. For instance, for Markov jump processes it is possible to find a synchronisation-like phenomenon where trajectories tend to exactly follow each other with a time delay. In the context of SDES, it is possible that the system exhibits noise-induced sychronisation or even noise-induced chaos. We explore both concepts in a stochastic Brusselator, where in the absence of noise the chemical system may exhibit self-sustained oscillations due to a Hopf bifurcation.
This is joint work with Maximilian Engel (FUB), Nathalie Unger (ZIB), Steffanie Winkelmann (ZIB), and Robin Chemnitz (FUB).
Dia: Dijous, 6 de juliol de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Luke Peterson (Univ. Colorado)
Títol: Dynamics Near the Triangular Points of the Sun-Earth-Moon Hill Restricted 4-Body Problem
Resum: The study of dynamics near the triangular points (L4 and L5) of 3-body systems within the Solar system has been well-documented in celestial mechanics due to their stability, e.g., the Trojan asteroids in the Sun-Jupiter system. However, these regions can become unstable in planet-moon systems once the gravitational effect of the Sun is incorporated (and, hence, made into a 4-body problem). The dynamics in the region near the Earth-Moon triangular points is interesting to astronomers (describing dynamical mechanisms to explain the lack of objects observed) and astrodynamicists (placing communications satellites for orbits near the Moon).
In this talk, we use the Hill restricted 4-body problem (HR4BP), a $\pi$-periodic time-dependent Hamiltonian dynamical system that is both a generalization of the circular restricted 3-body problem (CR3BP) and Hill's problem, to model the motion of a particle under the mutual gravitation of the Sun, Earth, and Moon in a coherent way.
In the HR4BP, the equilibrium point L4 (and, by symmetry, L5) of the CR3BP is replaced by a $\pi$-periodic orbit, which we call the dynamical equivalent of L4 (DE L4). Additionally, due to the periodicity of the model, we retain a 2:1 subharmonic periodic orbit about L4.
Now, on the one hand, we use semi-analytical methods (e.g., reduction to the center manifold) to seek an understanding of the local dynamics near DE L4; on the other hand, we use a flow map method to compute families of quasi-periodic invariant tori (and their stability) outside the radius of convergence of the center manifold reduction.
As this work is ongoing, we present preliminary results of the implemented techniques. Namely: the ineffectiveness of center manifold reduction at L4 (and what we can learn from this); the five families of 2-dimensional Lyapunov tori near DE L4 & the elliptic 2:1 resonant periodic orbit; the stability of invariant tori around DE L4; and, finally, bifurcations are discussed.
Comparisons will be made to the Bicircular restricted 4-body problem where applicable.
Dia: Dimecres, 12 de juliol de 2023
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Joan Carles Tatjer (UB)
Títol: Non smooth pitchfork bifurcations for a quasiperiodically forced piecewise linear map
Resum: We study a class of one-dimensional family of quasiperiodically forced maps F_{a,b}(x,theta)=(f_{a,b}(x,theta),theta+omega), where x is real, theta is an angle, and omega is an irrational frequency, such that f_{a,b}(x,theta) is a real piecewise linear map with respect to x of certain kind. For these families, we can prove the existence of a non-smooth pitchfork bifurcation. This means that there exists a map $b=b_0(a)$ such that
a) For b<b_0(a), f_{a,b} has two continuous attracting invariant curves and one continuous repelling one.
b) For b=b_0(a) it has one continuous repelling invariant curve and two semicontinuous (non-continuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles.
c) For b>b_0(a) it has one continuous attracting invariant curve.
This family is a simplfied version of the smooth family G_{a,b}(x,theta)=(arctan(ax)+b sin(theta),theta+omega) for which it seems that we have smooth and non-smooth pitchfork bifurcations.
This is a joint work with Àngel Jorba and Yuan Zhang.
Last updated: Friday, 28-Jul-2023 23:23:08 CEST