The One World Dynamics seminar is part of the One World (OW) seminar series pioneered by the One World seminars in probability. The aim of the One World Dynamics seminar is to cover the breadth of Dynamical Systems. This includes, but is not limited to,
Ergodic systems,
Hamiltonian systems,
Complex dynamics and low-dimensional maps,
PDE dynamics and infinite-dimensional systems,
Multiscale dynamics,
Stochastic and non-autonomous systems,
Network dynamics.
The objective of the seminar is to describe recent developments in the field of dynamics, but also to provide a platform that allows an exchange of expertise between different branches of this field. Every edition of the OWDS includes two talks; the first is open not only to researchers from all over the world, but also to Master students and PhD students from varied backgrounds.
Please subscribe to our mailing list by filling this form (note this is a two-step process, where a link to confirm the registration will be sent to by email. Please check you spam filter if you do not receive such an email). Seminars will take the format of Zoom Webinars. Approximately 15 minutes prior to the beginning of the lecture, a zoom link will be provided via the mailing list.
One World Dynamics seminars will run on the second Friday of each month, starting 11 September 2020.
Each session comprises of two talks. The first talk is aimed at a wide audience in the community of dynamical systems, setting the scene for the second talk, which can be more specialised. Each talk will last 40min, followed by 5min for discussions.
The first talk will start at 3pm Berlin time (2pm London, 9am Washington DC); click here for conversion in your local time.
The second talk will start at 3:45pm Berlin time (2:45pm London, 9:45am Washington DC).
The next One World Dynamics Seminar will take place on Friday 5 July 2024, 3pm Berlin time (2pm London, 9am Washington DC) - 4:30pm Berlin time (3:30pm London, 10:30am Washington DC) with talks by Christian Bick and Davide Sclosa.
Christian Bick, Vrije Universiteit Amsterdam, The Netherlands. Dynamical Systems on Graph Limits and Beyond. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Graphs capture interactions in networks of coupled dynamical systems. Since many networks relevant in real-world applications consist of many nodes, graph limits and dynamics thereon have become a convenient tool to understand the collective dynamics of network dynamical systems with infinitely many nodes. We give a brief overview of graph limits as well as network dynamical systems on such limits - in particular their relationship with dynamics on a graph with a large but finite number of nodes. We also review some explicit examples of dynamics on graph limits and generalizations.
Davide Sclosa, Vrije Universiteit Amsterdam, The Netherlands. Symmetries of Dynamical Systems on Graph Limits. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Network dynamical systems typically inherit the symmetries of their underlying graph. Such symmetries can shape the dynamics in an essential way, for example by inducing synchrony patterns or by facilitating the emergence of structurally stable heteroclinic cycles. We discuss how, analogous to the case of finite graphs, symmetries shape dynamics on graph limits. Moreover, we show that symmetries of a graph limit can influence the dynamics on large but finite graphs, even if these finite graphs are asymmetric.
Attention: we have moved to a standard Zoom seminar platform (no more Zoom webinar platform).
Zoom link : https://gatech.zoom.us/j/92713473429?pwd=Yk9vWllaNjVrNDV4UnVWbVZiL1JFQT09
Meeting ID: 927 1347 3429
Passcode: OWDS
Montie Avery, Boston University
Alex Blumenthal, Georgia Tech
Maximilian Engel, Freie Universität Berlin
Jessica Massetti, Università Roma Tor Vergata
Michela Ottobre, Heriot-Watt University
Jasmin Raissy, Université de Bordeaux
Matteo Tanzi, King's College London
Friday 10 May 2024
Carsten Hartmann, Cottbus University of Technology, Germany. Stochastic approximations: How can randomness emerge in a deterministic system? 3pm Berlin time (2pm London, 9am Washington DC). Abstract: This talk will give a high-level overview of stochastic approximations of deterministic dynamical systems with many degrees of freedom. Sometimes, such approximations can be rigorously derived from the central limit theorem, assuming for example that the initial conditions are randomly distributed. Mostly, however, stochastic approximations are used in a data-driven fashion to describe an intermediate regime between smooth deterministic dynamics, that describes only an idealised situation (and that is often the result of a model simplification or approximation), and a very complicated model (e.g. chaotic with unknown initial data) that may be closer to reality, but that is computationally infeasible or just too costly. I will discuss three prototypical examples of stochastic (more specifically: Gaussian) approximations: (1) the chemical Langevin equation of a chemical reaction network, (2) the Gaussian approximation of a small noise diffusion and (3) Hasselmann's approach to climate modelling and numerical weather prediction. The topic of the presentation is closely related to the second presentation by Georg Gottwald.
Georg Gottwald, University of Sydney, Australia. A stochastic approximation for the finite-size Kuramoto-Sakaguchi model. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: We perform a stochastic model reduction of the Kuramoto-Sakaguchi model for finitely many coupled phase oscillators with phase-frustration. Whereas coupled oscillators asymptotically exhibit stationary states and a constant order parameter, finite-size networks exhibit non-stationary fluctuations. We propose a stochastic description of the synchronized oscillators. This is achieved by approximating the collective effect of the non-entrained oscillators on the synchronized cluster as a Gaussian process. This allows for an effective closed evolution equations for the synchronized oscillators driven by a two-dimensional Ornstein-Uhlenbeck process. Our reduction reproduces the stochastic fluctuations of the order parameter and leads to a simple stochastic differential equation for the order parameter. This is joint work with Wenqi Yue.
Friday 12 April 2024
Núria Fagella, Universitat de Barcelona, Spain. 3pm Berlin time (2pm London, 9am Washington DC).
Anna Miriam Benini, Università di Parma, Italy. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC).
Wandering Domains and Non Autonomous Dynamics on the disk
Abstract: In one dimensional complex dynamics we have an increasingly detailed knowledge about stable components which are periodic and preperiodic. On the other hand, stable components which elude being (pre)periodic (aka wandering domains) also elude our full understanding and are currently an active topic of research. While much of the current research focuses on constructing examples showing a great variety of possibilities, in our work we propose an actual classification of wandering domains according to the behaviour of their internal orbits. This seamlessly leads us to analyzing nonautonomous dynamics for self-maps of the unit disk.
For autonomous iteration of inner functions (self-maps of the disk whose radial extension is a self map of the boundary a.e.) there is a remarkable dichotomy due to Aaronson, Doering and Mañé, according to which the internal dynamics of the map determines the dynamical properties of its boundary extension: either (almost all) boundary orbits converge to a single point, or (almost all) boundary orbits are dense.
In the nonautonomous setting the situation is more complicated. However, we present a generalization of this dichotomy which is, in a specific sense, optimal.
This is joint work with Vasso Evdoridou, Phil Rippon, and Gwyneth Stallard. Parts of this work are still in progress.
PDF of Núria Fagella's talk
PDF of Anna Miriam Benini's talk
Friday 9 February 2024
Vaughn Climenhaga, University of Houston, USA: Entropy as a dimension. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Various definitions of entropy can be used to quantify the complexity of a dynamical system. These are analogous to various definitions of dimension from fractal geometry; in this talk I will explore these connections and explain how tools from dimension theory can be used to construct dynamically relevant measures. In particular, I will describe joint work with Yakov Pesin and Agnieszka Zelerowicz in which we use these tools to construct the measure of maximal entropy, the Sinai-Ruelle-Bowen measure, and other equilibrium measures for uniformly hyperbolic (Anosov or Axiom A) systems.
Jason Day, University of Houston, USA: Measure of maximal entropy for dispersing billiards via dimension theory 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Dispersing billiards are examples of hyperbolic systems with singularities. Recently, it was shown by Viviane Baladi and Mark Demers that the measure of maximal entropy for the billiard map is unique for finite horizon dispersing billiard systems. We show an alternative approach to obtain this measure by modifying a dimension theoretic construction that was developed for uniformly hyperbolic systems. This talk will address how the billiard map is coded as a symbolic system and how the local product structure of the billiard map is used to create this measure. This is joint work with Vaughn Climenhaga.
Friday 10 November 2023
Marcel Guardia, Universitat de Barcelona, Barcelona, Spain: Unstable motions in Celestial Mechanics. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: One of the oldest problems in dynamical systems is the stability of the Solar System. That is, whether the Keplerian ellipses of planets' orbits in a planetary N-body problem undergo large long-term changes. Nowadays, it is known that the answer is rather nuanced and that stability and instability coexist for nearby initial conditions. In this talk I will explain how to construct unstable motions (chaos, oscillatory motions, and even Arnold diffusion) in different models in Celestial Mechanics.
Andrew Clarke, Universitat Politècnica de Catalunya, Barcelona, Spain: Why are inner planets not inclined? 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Consider the Newtonian 4-body problem, in the regime where 3 bodies (the planets) revolve on near-elliptical orbits around the other body (the sun). A long-held belief, culminating in the XVIII century in the first stability theorem of Laplace and Lagrange, is that the semimajor axes are stable. Assuming the initial conditions of the semimajor axes are of different orders, and that there is a large mutual inclination between planets 1 and 2, we prove that there are orbits of the 4-body problem where the semimajor axis of planet 3 can follow any itinerary, with arbitrary precision. In addition, along such orbits, we can make the eccentricity and inclination of the orbital ellipse of planet 2 follow any itinerary, again with arbitrary precision. For example, planet 2 may flip from prograde to retrograde nearly-horizontal revolutions. Moreover its orbital ellipse can go from near-circular to highly eccentric. This is an instance of the phenomenon known as Arnold diffusion.
Friday 13 October 2023
Mariana Haragus, Institut FEMTO-ST, Université de Franche-Comté, Besançon, France: Frequency combs in the Lugiato-Lefever model. Part 1: Bifurcation of periodic waves and spectral stability. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: The Lugiato-Lefever equation is a nonlinear Schrödinger-type equation with damping, detuning and driving, derived in nonlinear optics by Lugiato and Lefever (1987). While extensively studied in the physics literature, there are relatively few rigorous mathematical studies of this equation. Of particular interest for the physical problem is the dynamical behavior of frequency combs, which are optical signals consisting of a superposition of modes with equally spaced frequencies. Mathematically, these are stationary periodic, or localized, solutions of the Lugiato-Lefever equation. We focus on the periodic solutions bifurcating at the onset of Turing instability. Relying upon tools from local bifurcation theory we show the existence of periodic solutions and identify a class of stable solutions. We prove their spectral stability for general bounded perturbations using a Bloch decomposition and perturbation methods from the theory of linear operators.
Björn de Rijk, Department of Mathematics, Karlsruhe Institute of Technology, Germany: Frequency combs in the Lugiato-Lefever model. Part 2: Nonlinear dynamics and uniform subharmonic stability. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: The Lugiato-Lefever equation is a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. In recent years it has received considerable attention as a model for frequency combs generated by microresonators in periodic optical waveguides. One of the mathematical questions raised by the physical problem concerns the nonlinear stability of its periodic solutions. Typically, the circumference of the microresonator is a multiple of the period T of the solution so that it is natural to consider subharmonic, or NT-periodic, perturbations. For fixed positive integers N nonlinear stability against NT-periodic perturbations can be obtained with standard methods by exploiting the presence of a spectral gap. However, the size of the spectral gap, and thereby the allowable size of perturbations, goes to 0 as N -> infty, rendering the result not uniform in N. In this talk, I will show how uniform subharmonic stability can be obtained by following an approach which was originally developed to establish nonlinear stability of periodic patterns in reaction-diffusion systems against localized perturbations. The method relies on a delicate decomposition of the associated semigroup, spatio-temporal phase modulation and nonlinear damping estimates to control regularity. This is joint work with Mathew Johnson, Mariana Haragus and Wesley Perkins.
Friday 14 July 2023
Sebastian Riedel, FernUniversität in Hagen, Germany: Pathwise stochastic analysis and random dynamical systems. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Studying random dynamical systems generated by stochastic differential equations (SDEs) can be very helpful if one is interested in the long-time behaviour of the solution. A necessary condition to apply random dynamical systems is, however, that the solution to the SDE is pathwise defined. Although this is often the case in finite dimensional spaces, the picture is less clear when the solution process takes values in an infinite dimensional space. Rough paths theory provides a genuinely pathwise approach to study SDEs in finite and also infinite dimensional spaces. We discuss some insights the application of rough paths theory gave when studying dynamical properties of SDE solutions. In particular, we show how rough paths theory can be used to prove that certain singular stochastic delay differential equations induce random dynamical systems.
Mazyar Ghani Varzaneh, FernUniversität in Hagen, Germany: On the dynamics of the singular stochastic functional equations. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Stochastic functional equations often fail to be continuous with respect to the noise and the initial value. Therefore, there are only a few results about the dynamics of this family of equations. The main purpose of this talk is to set up a framework such that a flow property can be generated on (roughly speaking) a bundle of Banach spaces. In this talk, we state a modified version of the multiplicative ergodic theorem to generate the Lyapunov exponents. This, in particular, yields several results, including the existence of the stable and unstable manifold, which we briefly talk about them.
Friday 12 May 2023
Olivier Faugeras, Directeur de Recherche (Emeritus) at Inria, France : Two examples of thermodynamic limits in neuroscience. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: The human brain contains billions of neurones and glial cells that are tightly interconnected. Describing their electrical and chemical activity is mind-boggling hence the idea of studying the thermodynamic limit of the equations that describe these activities, i.e. to look at what happens when the number of cells grows arbitrarily large. It turns out that under reasonable hypotheses the number of equations to deal with drops down sharply from millions to a handful, albeit more complex. There are many different approaches to this which are usually called mean-field analyses. I present two mathematical methods to illustrate these approaches. They both enjoy the feature that they propagate chaos, a notion I connect to physiological measurements of the correlations between neuronal activities. In the first method, the limit equations can be read off the network equations and methods "à la Sznitman" can be used to prove convergence and propagation of chaos as in the case of a network of biologically plausible neurone models. The second method requires more sophisticated tools such as large deviations to identify the limit and do the rest of the job, as in the case of networks of Hopfield neurones such as those present in the trendy deep neural networks.
Romain Veltz, Inria Center of University Côte d’Azur, France: Study of a mean field of a network of 2d spiking neurons. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: In this talk, I will present some results regarding the dynamics of a network of stochastic spiking neurons akin to the "generalised linear model". This network is an elaboration of the one introduced in [De Masi et al. 2014] by generalising the dynamics of the individual neurons. This allows to capture most of the known intrinsic neuronal spiking, like bursting for example, and thus to study the effect of the neuron dynamics on the macroscopic one. The model presents some challenges. It is a nonlinear Piecewise Deterministic Markov process with explosive flow and unbounded total rate function. I will present some theoretical results regarding the linearised equation (well posedness, ergodicity) and will highlight the difficulties associated to the nonlinear one. In particular, I will prove the existence of invariant distributions for the nonlinear process. Some numerical applications will be provided.
Friday 10 March 2023
Konstantinos Spiliopoulos, Boston University, USA: Rate of homogenization for fully-coupled McKean-Vlasov SDEs. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest. Joint work with Zachary Bezemek.
Pavlos Zoubouloglou, University of North Carolina, USA: Large Deviations for Small Noise Diffusions Over Long Time. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan (1976) are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts.
Friday 10 February 2023
Arnd Scheel, University of Minnesota, USA: Propagation into unstable state: front selection criteria. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: I will present an overview of recent work on front invasion into unstable states. Originating in classical work for the Fisher-KPP equation, questions related to front invasion are central to our understanding of instabilities in spatially extended systems, ranging from fluids to problems in ecology and sociology. One observes how localized disturbances evolve into propagating fronts which in turn select a particular state in their wake. The challenge to theory is to predict both front speeds and selected states in the wake. It turns out that many predictions can be phrased in terms of a marginal stability conjecture. The talk will explain this conjecture, leading to a mathematically rigorous statement and proofs that will be explained in the second talk. I will also discuss variations, when assumptions fail or when assumptions can be weakened, including transitions to pushed fronts, spatio-temporal resonances, node conservation, and staged invasion.
Montie Avery, Boston University, USA: Universal selection of pulled fronts. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: I will present recent work which rigorously establishes selection of invasion fronts in reaction-diffusion systems under marginal spectral stability assumptions, as predicted by the marginal stability conjecture. Marginal spectral stability naturally leads to diffusive dynamics in the leading edge, so that steep initial conditions develop a diffusive tail which determines the invasion speed and the state selected in the wake. Front selection may then be phrased as a diffusive stability problem for an approximate solution which glues this diffusive tail to a selected front in the wake. The matching with the diffusive tail introduces terms in the perturbation equation which are supercritical from the point of view of diffusive decay, and overcoming this relies crucially on sharp decay estimates for the linearization about an invasion front. I will give an overview of the underlying selection mechanism, the resulting proof of front selection, and new methods for diffusive stability estimates developed in the course of the proof.
Friday 13 January 2023
Viktor Ginzburg, University of California, USA: Topological Entropy, Barcodes and Morse Theory. 5pm Berlin time (4pm London, 11am Washington DC). Abstract: Topological entropy is one of the fundamental invariants of a dynamical system, measuring its complexity. In this talk, we discuss a connection between the topological entropy of a Hamiltonian dynamical system, e.g., the geodesic flow or a Hamiltonian diffeomorphism, and the Morse homology filtered by the action and associated to it via a variational framework in a very broad sense. We introduce a new invariant associated with the system, the barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in low dimensions. The talk is based on joint work with Erman Cineli, Basak Gurel and Marco Mazzucchelli.
Basak Gurel, University of Central Florida, USA: Capturing Topological Entropy via Floer Theory: Tomographs and Crossing Energy. 5:45pm Berlin time (4:45pm London, 11:45am Washington DC). Abstract: In this talk we explore the relations between barcode entropy and topological entropy in detail for Hamiltonian diffeomorphisms and also for geodesic flows. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that in lower dimensions — two or three, depending on the setting — the barcode entropy is equal to the topological entropy. The talk is based on joint work with Erman Cineli, Viktor Ginzburg and Marco Mazzucchelli.
Friday 09 December 2022
Nils Berglund, Université d'Orléans, France: Noise-induced transitions between limit cycles. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: We are interested in quantifying the effect of weak noise on planar ODEs admitting two limit cycles, separated by an unstable periodic orbit. The main effect of the noise is to induce random transitions between neighbourhoods of the limit cycles. Obtaining sharp results on the distribution of these transition times is surprisingly difficult, mainly because these stochastic differential equations lack reversibility. In this talk, we will present sharp small-noise asymptotics on the expectation of the transition time, and give a description of their distribution, which reveals links to extreme-value theory. The results are partly based on joint work with Barbara Gentz (University of Bielefeld).
Maximilian Engel, Freie Universität Berlin, Germany: Shear-induced chaos via stochastic forcing: a tale of finding positive Lyapunov exponents. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: We discuss the phenomenon of shear-induced chaos, coined by Wang and Young about twenty years ago and referring to chaotic behavior as a result of shear being magnified by some forcing, in the context of stochastic perturbations. As a latest result, we show the positivity of Lyapunov exponents for the normal form of a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes a series of related results for simplified situations which we can exploit by studying suitable limits of the shear and noise parameters. Some general ideas concerning conditioned random dynamics, computer-assisted proofs and continuity of Lyapunov exponents will be highlighted along the way.
Friday 14 October 2022
Stephen Coombes, University of Nottingham, UK: Modelling large scale brain dynamics: An overview, mathematical challenges, and a new framework. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Neural mass models have been actively used since the 1970s to model the coarse-grained activity of large populations of neurons and synapses. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. In this talk I will discuss a simple spiking neuron network model that has recently been shown to admit to an exact mean-field description for synaptic interactions. This has many of the features of a neural mass model coupled to an additional dynamical equation that describes the evolution of population synchrony. I will show that this next generation neural mass model is ideally suited to understanding the patterns of brain activity that are ubiquitously seen in neuroimaging recordings. I will also touch upon open mathematical challenges in describing the link between structural- and functional-connectivity in large scale brain dynamics and how this might be understood using a phase-amplitude reduction. Time permitting, I will move on to consider extensions of this work to include a thalamic component. The thalamus is a body of neural cells that relays impulses to the cerebral cortex from the sensory pathways. Feedback from the cortex gives rise to thalamo-cortical loops that generate emergent brain rhythms from the interplay of single cell ionic currents and network mechanisms. The relevance of this extended model for understanding brain response to sensory drive will be illustrated with a comparison to human neuroimaging data for median nerve stimulation. A further mathematical challenge is to understand the Arnol'd tongue structure that this model generates.
Gemma Huguet, Universitat Politècnica de Catalunya, Spain: Oscillatory dynamics and communication in neuronal network models. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Oscillations are ubiquitous in the brain, but their role is not completely understood. In this talk we will focus on the study of oscillations in neuronal networks. I will introduce some neuronal models and I will show how tools from dynamical systems theory, such as the parameterization method for invariant manifolds, can be used to provide a thorough analysis of the oscillatory dynamics. I will show how the conclusions obtained may contribute to unveiling the role of oscillations in certain cognitive tasks.
Friday 09 September 2022
Jacob Bedrossian, University of Maryland, USA: Lower bounds on the top Lyapunov exponent of stochastic systems. 2pm Berlin time (1pm London, 8am Washington DC). Abstract: We review our recent joint work with Alex Blumenthal and Sam Punshon-Smith, which introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for either deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called “projective process”); and (B) an L1 -based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. We will briefly review this method and end with a discussion of some open problems. Further details are discussed in Sam Punshon-Smith's talk.
Sam Punshon-Smith,Tulane University, USA: Positive Lyapunov exponent for 2d Stochastic Galerkin Navier-Stokes. 2:45pm Berlin time (1:45pm London, 8:45am Washington DC). Abstract: In this talk I will discuss more details about our recent proof of positivity of the top Lyapunov exponent for Galerkin truncations of the 2d Stochastic Navier-Stokes equations using our method outlined in the previous talk by Jacob Bedrossian. First, I will explain aspects of the proof of our L1-type hypoelliptic estimate on the Fisher information and describe how positivity of the top Lyapunov exponent for 2d Galerkin-Navier-Stokes then comes down to verifying hypoellipticity of its lift to the projective bundle (or more generally on the principle SL_N bundle). I will then explain our recent proof with Jacob Bedrossian verifying this condition for the Stochastic Galerkin Navier-Stokes equations using techniques from computational algebraic geometry.
Friday 08 July 2022
Tomás Caraballo, Universidad de Sevilla, Spain: An introduction to the theory of pullback attractors for non-autonomous/random dynamical systems with applications
3pm Berlin time (2pm London, 9am Washington DC). Abstract: The theory of pullback attractors for Random/Non-autonomous Dynamical Systems has been extensively developed over the last three decades, mainly due to the fact that it provides a suitable framework to analyze the global behavior of dynamical systems containing some kind of stochasticity or randomness as well as some time dependent forcing. In this talk we will introduce the basic tools of the theory of pullback attractors for handling both non-autonomous and random dynamical systems. We will show how both problems can be analyzed in a unified formulation thanks to the concept of cocycle. We will also emphasize on the different effects that different kind of noise can produce on the asymptotic behaviour of the solutions. Our results will be illustrated with some basic academic examples as well as some other interesting models appearing in the applied sciences (for example, reaction-diffusion equations and chemostast models).
Thorsten Hüls, Universität Bielefeld, Germany: Angular Values of Linear Discrete Time Dynamical Systems. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: In this presentation we introduce the new notion of angular values for a nonautonomous or random linear dynamical system. The angular value of dimension s measures the maximal average rotation which an s-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. This notion works for subspaces of arbitrary dimension and avoids to specify orientation. Our main results relate the notion of angular values to the well-known dichotomy (or Sacker-Sell) spectrum and its associated spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over the whole Grassmannian, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. On the one hand this reduction leads to explicit formulas in the autonomous case. On the other hand it serves as the basis for an algorithm which allows to compute angular values of arbitrary dimension. We demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics. (Joint work with Wolf-Jürgen Beyn and Gary Froyland.)
Friday 10 June 2022
Greg Pavliotis, Imperial College London, UK: Phase Transitions, Logarithmic Sobolev Inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions.
3pm Berlin time (2pm London, 9am Washington DC). Abstract: We present recent results on the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. In particular, we explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-particle system and the presence or absence of phase transitions for the mean field limit. The non-degeneracy of the LSI constant is shown to have far reaching consequences, especially in the context of uniform-in-time propagation of chaos and the behaviour of equilibrium fluctuations. Our results extend previous results related to unbounded spin systems and recent results on propagation of chaos using novel coupling methods. This is joint work with Matias Delgadino, Rishabh Gvalani and Scott Smith.
Anastasia Borovykh, Warwick University, UK: On efficient distributed optimization with mirror maps. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: The choice of mirror map has a significant impact on both the theoretical and numerical performance of the Mirror Descent (MD) algorithm. Compared to Projected Gradient Descent (PGD), with the right choice of mirror map, MD can converge much faster to the optimum. While there exists no theoretical or algorithmic framework to explain how to compute an optimal mirror map for a given problem, mirror maps for some particular classes of problems are well known. Mirror descent, and especially the effect of the choice of the mirror map for distributed optimization problems has received much less attention. Distributed optimization problems, even when otherwise unconstrained, have to satisfy a consensus constraint, and existing algorithms do not capture the geometry of the consensus manifold. Motivated by the attractive features of the mirror descent algorithm described above, we attempt to first answer the question:1) Does there exist a distributed variant of Mirror Descent that can accurately capture the geometry of distributed optimization models? We first define a natural generalisation of the standard distributed optimization algorithm to the MD setting. However - as we will show, this first formulation (which we call Distributed Mirror Descent, DMD) does not converge to the unique solution of the problem. The second question we seek to address in this talk is: 2) How should the dynamics of distributed mirror descent be modified, so that convergence to the exact solution is guaranteed? We consider the more general case of stochastic dynamics with additive noise, where the noise is added to account for corrupted gradient information, data sub-sampling or errors due to the network We will show that it is possible to formulate an Exact DMD algorithm which does allow for a closer convergence to the optimum than the standard DMD algorithm.
Friday 11 March 2022:
Matthew Foreman, University of California, Irvine: **4pm** Berlin time (3pm London, 10am Washington DC).
Please note the slightly different start time and check your local time zone.
Philipp Kunde, Universität Hamburg: **4:45pm** Berlin time (3:45pm London, 10:45am Washington DC).
Please note the slightly different start time and check your local time zone.
Title and abstract are for both talks. Title: The complexity of the Structure and Classification of Dynamical Systems. Abstract: Some classical programs in both smooth dynamical systems and ergodic theory are infeasible using inherently countable information. These talks discuss what this means, and how to measure of the complexity of classification problems using Borel reducibility and benchmarks.
Friday 11 February 2022:
Viviane Baladi, LPSM Sorbonne Université Paris: Studying Sinai billiards via anisotropic spaces . 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Sinai billiards (or the periodic Lorentz gas) are natural dynamical systems which have been challenging mathematicians for half a century. In the past decade, a new mathematical tool to study them has emerged : Ruelle transfer operators acting on scales of anisotropic Banach spaces. This tool was introduced in 2011 in the billiards setting by Mark Demers and Hong-Kun Zhang, who gave a new proof of Lai-Sang Young's celebrated 1998 result of exponential mixing for the SRB measure of the billiard map. I will survey rigorous results obtained on dispersing billiards (for both discrete and continuous time) using this approach, leading to recent work with M. Demers on the measure of maximal entropy and more general Gibbs states for the billiard map.
Mark F. Demers, Fairfield University, Connecticut: Projective Cones for Dispersing Billiards . 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: We describe the recent construction of Birkhoff cones which are contracted by the action of transfer operators corresponding to dispersing billiard maps. The explicit contraction provided by this construction permits the study of statistical properties of a variety of sequential and open billiards. We will discuss some applications of this technique to chaotic scattering and the random Lorentz gas. This is joint work with C. Liverani.
Friday 14 January 2022:
Christopher Jones, University of North Carolina (UNC-CH), USA: Generalizing Stability Indices for Nonlinear Waves. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: The Maslov Index offers a way to calculate the number of (real) unstable eigenvalues for the linearized operator at a nonlinear wave, but it requires some special structure of the underlying system, namely that the eigenvalue equations be Hamiltonian, or at least transformable to Hamiltonian. Since many systems of interest do not have this property, a natural question is whether such an index can be formulated more generally. The most tractable situation is when the underlying PDE is a system of two equations. The generalization to arbitrary dimensions is more involved, but the ideas can be expressed clearly for a system of two equations using the right coordinates. What emerges is perhaps surprising in that it leads to an index that both generalizes and complements the Maslov Index. The goal of this lecture will be to explain this somewhat ironic statement through presenting the way we thought about this issue.
Graham Cox, Memorial University, Canada: A Generalized Maslov Index for Non-Hamiltonian Systems. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Elaborating on the previous talk, I will explain how to define a generalized Maslov index that has desirable monotonicity properties for reaction–diffusion systems. Computation of this index is greatly simplified if a certain invariance condition holds. (For the classical Maslov index the desired invariance is guaranteed by the Hamiltonian structure of the equation.) For our more general index we cannot guarantee that the corresponding invariance condition holds, but this turns out to be a feature, rather than a shortcoming, of the theory. For instance, I will show that the Turing instability occurs precisely when this invariance fails, and hence the generalized index is non-zero.
These talks represent joint work of the speakers with Thomas Baird, Paul Cornwell and Robert Marangell.
Friday 10 December 2021:
Peter Ashwin, University of Exeter, United Kingdom: Heteroclinic dynamics: From networks in physical space to networks in phase space. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Orbits that limit to different invariant sets in the past and future have long been appreciated as dynamical structures that help organize more complicated dynamics at their bifurcation, and for example as structures that enable one to characterise fronts in spatio-temporal dynamics. However, they can also be very helpful to understand input-dependent dynamics, especially for coupled systems of near-identical units, where robust heteroclinic attractors may arise due for example to symmetries. Attracting networks made of heteroclinic orbits provide examples that challenge our assumptions about how typical attractors behave. In this talk I aim to give a flavour of some of the theoretical results, applications and problems in this area.
Alexandre Rodrigues, Universidade do Porto, Portugal: Rank-one strange attractors versus Heteroclinic tangles. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: In this seminar, we present a mechanism for the emergence of strange attractors (observable chaos) in a two-parametric periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. These relative positions may be determined using the Melnikov method. Generalising [1], we prove the existence of many complicated dynamical objects in the two-parametric family, ranging from rank-one attractors to Hénon-type strange attractors. We also draw a plausible bifurcation diagram associated to the problem under consideration and we show that the "occurrence of heteroclinic tangles" is a prevalent phenomenon. [1] I. Labouriau, A. Rodrigues, Periodic forcing of a heteroclinic network, Journal of Dynamics and Differential Equations 2021 (to appear), DOI 10.1007/s10884-021-10054-w
Friday 8 October 2021: Stochastic dynamics
Vadim Kaloshin, Institute of Science and Technology, Austria: Deformational Spectral Rigidity of analytic Bunimovich stadia and Bunimovich squashes. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: Consider a convex domain on the plane and the associated billiard inside. The length spectrum is the closure of the union of perimeters of all period orbits. The length spectrum is closely related to the Laplace spectrum, through the wave trace and the well-known question: ``Can you hear the shape of a drum?'' A domain is called dynamically spectrally rigid (DSR) if any smooth deformation preserving the length spectrum is an isometry. We show that an analytic deformation of a Bunimovich stadium (resp. a Bunimovich squash) is an isometry. One can naturally talk about DSR for geodesic flows. This result can be viewed as an analog of a well-known result by Guillemin-Kazhdan that a geodesic flow on a compact negatively curved surface is DSR. This is a joint work with J. Chen and H. Zhang.
Hamid Hezari, UC Irvine, USA: The inverse spectral problem for nearly circular domains and nearly circular ellipses. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: The so called "Poisson relation" is the link between the length spectrum and Laplace spectrum of a bounded smooth domain. We will use this connection to present a proof that ellipses of small eccentricity are spectrally unique among all smooth domains. We will also show that nearly circular domains with one axial symmetry are Laplace spectrally rigid within this class. This part uses the result of De Simoi-Kaloshin-Wei on the length spectral rigidity of such domains.
Friday 10 September 2021: Stochastic dynamics
Björn Schmalfuß, Friedrich-Schiller-Universität Jena, Germany: Random dynamical systems - a short overview. 3pm Berlin time (2pm London, 9am Washington DC). Abstract
Luigi Amadeo Bianchi, Università di Trento, Italy: Modulation equations and SPDEs. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: We consider the impact of additive space-time white noise on a pitchfork bifurcation of the stochastic Swift–Hohenberg equation with polynomial nonlinearity in an unbounded domain. We use an approximation via modulation equations, improving the standard methods to address the weak regularity of solutions. This approximation gives a useful tool to analyse how the noise influences the dynamics close to a change of stability.
Friday 9 July 2021: Transfer operators
Gary Froyland, University of New South Wales, Australia: Learning geometry and statistics from the spectrum of time-dependent dynamics. 2pm Berlin time (1pm London, 8am Washington DC). Abstract: I will discuss spectral approaches to statistical laws and mixing geometry for nonlinear dynamics, before introducing random or nonautonomous dynamical systems and describing recent extensions of spectral methods to the time-dependent setting. Applications to geophysical fluid flows will be also be presented.
Cecilia Gonzalez-Tokman, University of Queensland, Australia: Lyapunov exponents for transfer operator cocycles of random one-dimensional maps. 2:45pm Berlin time (1:45pm London, 8:45am Washington DC). Non-autonomous or random dynamical systems provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. In the last fifteen years, transfer operators have been combined with multiplicative ergodic theory to shed light on ergodic-theoretic properties of random dynamical systems, through the so-called Lyapunov-Oseledets spectrum. While the scope of this framework is broad, in practice it is challenging to identify and even approximate this spectrum. In this talk, we present examples of one-dimensional map cocycles where we are able to describe the Lyapunov-Oseledets spectrum and understand some of its stability properties under perturbation. This talk is based on joint works with Gary Froyland and Anthony Quas.
Friday 11 June 2021: Billards
Carlangelo Liverani, Università di Roma Tor Vergata, Italy: Statistical properties of hyperbolic Billiards. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: The study of the strong statistical properties of billiards (e.g. the rate of mixing) has seen a rapid evolution in the last 25 years and many new ideas have been developed to further their study.
Yet, many problems remain wide open. In particular, next to nothing is known about systems of many balls or about aperiodic extended systems (e.g. Random Lorentz gasses).
Nevertheless, some significant progresses have been achieved in the last years that cast some hope for further developments.
I will describe some aspect of such a state of affairs.
Jacopo De Simoi, University of Toronto, Canada. Dynamical rigidity of convex Billiards. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Convex billiards are a classical topic in conservative dynamics. Typically, their dynamics is qualitatively very intricate, since it showcases a coexistence of hyperbolic dynamics and KAM phenomena. Understanding long-term statistical properties of the dynamics with the current technology is essentially an intractable problem. Here I venture in the opposite direction and I will discuss dynamical inverse problems: how much geometrical information can be extracted from the dynamics? More precisely: what can be deduced about the billiard table if one knows the lengths of all periodic orbits? The quantum version of this question has been famously stated as "Can one hear the shape of a drum?" In this talk I will review the latest results and describe the next steps in this direction. This is a joint project with Vadim Kaloshin.
Friday 12 March 2021: Network dynamics
Mike Field, University of Santa Barbara, USA: Models for network dynamics: Function, Evolutionary Bifurcation, and non-convex Optimization. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: We start with a discussion of two questions related to the study of dynamics on a network: How do we define network dynamics, from the point of view of dynamical systems theory? What is a (good) “toy model”?
We then specialize to discuss a possible toy model for evolutionary bifurcation. The origins of the model lie in the study of non-convex optimization on a 2 layer neural net but this part of the talk will not be about neural nets. Rather, we discuss dynamics, which is gradient but very far from simple, symmetric bifurcation and some of the analytic tools. The underlying focus will be on gaining some insight into the non-convex optimization which we believe may have a role in evolutionary bifurcation. All of this will be discussed entirely within the framework of dynamical systems, networks with a function, and bifurcation: no prior knowledge of neural nets or machine learning is needed. Methods involve real analytic analysis and some ideas from equivariant bifurcation.
Much of the research described in the talk arises from a collaboration with Yossi Arjevani (NYU and Hebrew University of Jerusalem, from July).
Eddie Nijholt, University of Sao Paulo, Brazil: Exotic symmetry in networks. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract : All throughout science and engineering, there is an abundance of phenomena that are modelled by network dynamical systems. Despite this prevalence, it remains unclear precisely how network structure impacts the dynamics. Some important results in this direction rely on the observation that a symmetry in a network translates to a symmetry of the dynamical system. However, most networks need not have any symmetry, and many other network properties (such as node-dependency, feed-forward structure, synchrony spaces, hidden symmetry and so forth) are still known to impact the dynamics. We will see that most of these features can still be captured as symmetry, provided one allows for more `exotic symmetries'. Examples include semigroups, categories and, most recently, quiver symmetry. For certain types of interaction structure, the network topology itself is even equivalent to the presence of such a generalised symmetry! As an important consequence, this connection allows us to preserve network structure in most reduction techniques, which in turn makes it possible to analyse bifurcations in such systems.
Friday 12 February 2021: Hamiltonian dynamics
Susanna Terracini, Università di Torino, Italy: On the variational approach to the N-body problem. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: In its full generality, the N-body problem of Celestial Mechanics has challenged many generations of mathematicians.
It is commonly accepted, since the early works by H. Poincar\'e, that the periodic problem, through its associated action spectrum, carries precious information on the whole dynamics of a Hamiltonian system. Therefore, the problem of the existence and the qualitative properties of periodic and other selected trajectoriesfor the N-body problem (from the classical celestial mechanics point of view to more recent advances in molecular and quantum models) has been extensively studied over the decades, and, more recently, new tools and approaches have given a significant boost to the field.
We shall review some old an new results on the existence and classification of selected trajectories of the classical N-centre and N-body problem, withan emphasis on new analytical and geometrical techniques.
Andrea Venturelli, Université d´Avignon et des Pays du Vaucluse, France: Hyperbolic motion in the Newtonian N-body problem with arbitrary limit shape. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract : We prove for the N-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level h>0 of the motion can also be chosen arbitrarily. Our approach is based on the construction of a global viscosity solutions for the Hamilton-Jacobi equation H(x,du(x))=h. Our hyperbolic motion is in fact a calibrating curve of the viscosity solution. The presented results can also be viewed as a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits. It is a joint work with Ezequiel Maderna.
Friday 8 January 2021: Stochastic dynamics
Martin Hairer, Imperial College, London, United Kingdom: Averaging of non-Markovian SDEs. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter H > 1/2. We show that unlike in the case H = 1/2, convergence to the averaged solution takes place in probability and the limiting process solves the "naïvely" averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma. This is joint work with Xue-Mei Li.
James MacLaurin, New Jersey Institute of Technology, United States : An Emergent Autonomous Flow for Spin Glass Dynamics. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: We study the large size limiting dynamics of symmetric and asymmetric spin-glass models of size N. These are stochastic interacting particle systems with random connections, of zero mean, and relatively high variance. They are generally considered to be prototypes of high dimensional stochastic systems navigating a complex disordered environment, with diverse applications, including magnetism, colloids, neuroscience, and deep-learning algorithms. Existing work on the large size limiting dynamics has established delayed integro-differential equations that are very difficult to analyze rigorously. The content of this work is to derive an emergent flow operator that is autonomous, and hence more amenable to standard PDE techniques. It is proved that the same flow operator holds for a wide range of initial conditions, including when the distribution of the initial values of the spins depend on the realization of the connections. In particular, the flow operator is accurate for chaotic initial conditions (a "deep quench"), and initial conditions given by the long-time equilibrium Gibbs measure. The analysis is in terms of the double empirical process: this contains both the spins, and the field felt by each spin, at a particular time (without any knowledge of the correlation history), spanning M "replicas" with identical connections and independent stochasticity. Preliminary numerical results suggest that, as the temperature is lowered, a "dynamical phase transition" occurs when a stable fixed point of this flow destabilizes (as long as there are more than two replicas).
Friday 11 December 2020: Ergodic systems
Lai-Sang Young, Courant Institute, New York, United States: Observable events and typical trajectories in dynamical systems. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: I will present ideas related to "typical solutions" for finite and infinite dimensional dynamical systems, deterministic or stochastic. In finite dimensions, one often equates observable events with positive Lebesgue measure sets, and view as physically relevant invariant measures that reflect large-time behaviors of positive Lebesgue measure sets of initial conditions. I will review basic concepts, and propose a simple picture that one might hope to be true. Reality can be messy for deterministic systems, unfortunately, but the addition of a small amount of random noise will bring this picture about. I will finish with some results that point to a notion of "typical solutions" for certain infinite dimensional systems. These ideas will be discussed in greater generality in the second talk today by Alex Blumenthal.
Alex Blumenthal, Georgia Tech, United States: SRB measures and visbility for infinite dimensional dynamical systems with an eye towards potential applications to PDE. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: I will talk about the extension to the setting of Banach space mappings a concept which has proven highly useful in the study of finite-dimensional dynamical systems exhibiting chaotic behavior, that of SRB measures. This extended notion of SRB measure and our results potentially apply to a large class of dissipative PDE, including dissipative parabolic and dispersive wave equations. We generalize two results known in the finite-dimensional setting. The first is a geometric result, absolute continuity of the stable foliation, which in particular implies that an SRB measure with no zero exponents is visible, in the sense of time averages converging to spatial averages, with respect to a large subset of phase space. The second is the characterization of the SRB property in terms of the relationship between a priori different quantifications of chaotic behavior, Lyapunov exponents and metric entropy. This work is joint with Lai-Sang Young.
Friday 13 November 2020: Multiscale dynamical systems
This month's seminars are devoted to multiscale dynamics. For evolutionary equations with multiple scales, the main goal is the construction of effective equations that describe the limiting dynamics.
To achieve this goal, there are various methods, depending on the structure of the underlying system. In the past decades, so-called gradient flow structures and rate independent evolution equations have been a focus of activity. The first talk will discuss gradient flow structures, and rate-independent evolution is a central theme of the second talk. Applications of these structures will be discussed in the talks.
Alexander Mielke, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany: Convergence for multiscale gradient systems with applications to fast-slow reaction systems. 3pm Berlin time (2pm London, 9am Washington DC). Abstract: For gradient-flow equations, the dynamics is driven by the interaction of the energy landscape and the dissipation structure. Both can generate independent spatial or temporal microstructure. Based on the energy-dissipation principle a construction of effective equations will be presented that generalizes the theory of Sandier-Serfaty from 2004.
Various applications will be treated, like homogenization and fast-slow reaction systems.
Riccarda Rossi, University of Brescia, Italy: Balanced viscosity solutions to (multi)rate-independent systems. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC). Abstract: Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z, as will be explained in the talk. Over the last 15 years, the vanishing-viscosity approach to the mathematical modeling of this kind of processes has been developed as a method to select solutions with a mechanically feasible behavior at jumps.
We aim to extend this approach, regularizing both the static equation and the rate-independent flow rule by adding viscous dissipation terms modulated by coefficients that vanish to zero with different rates. Thus, the displacement and the internal variable relax to elastic equilibrium and to rate-independent evolution, respectively, with different relaxation rates.
We will illustrate how this vanishing-viscosity analysis leads to a notion of Balanced Viscosity solution to the original rate-independent system that provides an accurate description of the system behavior at jumps.
Friday 9 October 2020: The Mandelbrot Set
This month’s session is dedicated to complex and low-dimensional dynamics, and will focus specifically on holomorphic dynamics in one variable. We consider dynamical systems on the Riemann sphere whose evolution rule is given by a complex polynomial. Here infinity is a (super-)attracting fixed point, and the filled Julia set is the set of points with bounded orbit. The quadratic family, P_c(z)=z^2+c, gives rise to the Mandelbrot set: the set of parameters c such that the filled Julia set of P_c is connected. The Mandelbrot set is one of the central objects in one-dimensional holomorphic dynamics; this week’s talks both discuss recent progress in its study, in a manner accessible to a general dynamical systems audience.
Misha Lyubich, Stony Brook, USA: On the MLC Conjecture. 3pm Berlin time (2pm London, 9am Washington DC, 9pm Beijing). Abstract: The MLC Conjecture, on local connectivity of the Mandelbrot set, is one of the central open problems in Holomorphic Dynamics. It turned out to be intimately related to many other geometric themes of the area: Fatou's Conjecture on the density of hyperbolicity, self-similarity features of the Mandelbrot set, Lebesgue measure of Julia sets, and so on. We will give an introduction to this problem and an update on its current status. Video (external link)
Luna Lomonaco, Instituto Nacional de Matemática Pura e Aplicada, Brasil: The Mandelbrot set and its satellite copies. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC, 9:45pm Beijing). Abstract: Computer experiments reveal the existence of small homeomorphic copies of the Mandelbrot set M inside itself; the existence of such copies was proved by Douady and Hubbard. Each little copy is either primitive (with a cusp on the boundary of its main cardioid region) or a satellite (without a cusp). Lyubich proved that the primitive copies of M satisfy a stronger regularity condition: they are quasiconformally homeomorphic to M. The satellite copies are not quasiconformally homeomorphic to M (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In joint work with C. Petersen we prove that the answer is negative in general. Video (external link)
Friday 11 September 2020: PDE dynamics and infinite-dimensional systems
Thierry Gallay, Université Grenoble Alpes, France: Dynamical systems and dissipative PDEs on unbounded domains. 3pm Berlin time (2pm London, 9am Washington DC, 9pm Beijing). Abstract: Dynamical systems associated with dissipative partial differential equations on unbounded spatial domains can exhibit interesting and rather unusual phenomena, which are related to noncompact symmetry groups, infinite-dimensional phase space, or infinite number of effective degrees of freedom. The aim of this introductory talk is to discuss a few examples originating from reaction-diffusion systems or incompressible fluid mechanics. Special attention will be paid to the dynamical relevance of scaling invariance, to the unclear phenomenon of nonlinear stabilization, and to the local convergence towards equilibria in extended dissipative systems.
Gabriela Jaramillo, University of Houston, USA: A normal form for rotating wave solutions in oscillatory media with nonlocal coupling. 3:45pm Berlin time (2:45pm London, 9:45am Washington DC, 9:45pm Beijing). Abstract: We consider an abstract oscillating chemical reaction with a fast component that can be eliminated adiabatically. As a result, this fast variable can be viewed as a nonlocal diffusive process, and one can model this reaction using a system of integro-differential equations. However, the nonlocal character of the equations prevents the use of standard techniques from spatial dynamics, like construction of a center manifold and the use of normal form theory, to study existence of spiral waves. We therefore turn to methods from functional analysis and perturbation theory to study this system. In particular, we derive Fredholm properties for the relevant convolution operators and use Lyapunov-Schmidt reduction to arrive at a normal form reminiscent of the complex Ginzburg-Landau equation.