Dynamical Systems Seminars

Spring 2013

The Dynamical Systems seminar is held on Monday afternoon at 4:00 PM in MCS 148. Tea beforehand at 3:45 PM in MCS 153.

- January 28
Enrique Thomann (Oregon State)

Title: Probabilistic methods in some nonlinear partial differential equations.

Abstract: The deep connections between stochastic processes and parabolic evolution equations serve as an example of the interaction between analysis and probability. This connections provides sometime alternative formulation of questions and methods of study that enrich our understanding of the problems and of the methods.In this talk I will survey results illustrating these connections in the context of some nonlinear equations. I will use basic tools of probability theory to obtain representations of solutions of some nonlinear equations, including most notably the Navier-Stokes equations, in terms of a multiplicative functional defined on a branching processes. The talk will include simpler examples to motivate the structures needed for the more general case. The presentation is based on work conducted with colleagues at Oregon State University and at University of Arizona. - February 4 Nathan Kutz (Washington)

Title: Sparse sensing and machine learning strategies for characterizing complex dynamical systems

Abstract: Sparse sensing is used to determine the flow characteristics around a cylinder (Reynolds number and pressure/flow field) from a sparse number of pressure measurements on the cylinder. Using a supervised machine learning strategy, library elements encoding the dimensionally reduced dynamics are computed for various Reynolds numbers. The use of convex $L^1$ optimization is then used with a limited number of pressure measurements on the cylinder to reconstruct the full pressure field and the resulting flow field around the cylinder. Aside from the highly turbulent regime (large Reynold's number) where only the Reynold's number can be identified, accurate reconstruction of the pressure field, flow field and Reynold's number are achieved. The combination of dimensionality reduction, sparse sensing, and machine learning can be broadly applied to characterizing complex dynamical systems through limited measurements and/or sensors. - February 11 Nitsan
Ben-Gal (Institute for Mathematics and its Applications, Minnesota)

Title: Asymmetric Grow-Up Equations and their Non-Compact Global Attractors

Abstract: In this talk we will discuss recent developments in the study of asymptotically asymmetric slowly non-dissipative reaction-diffusion equations, also known as grow-up equations. We will particularly focus on the long-time behavior of solutions and the non-compact global attractor structure for these PDEs. I will present the background of this problem, and introduce recent research utilizing nymerical continuation methods and numerous geometrical methods, including Fucik spectra, nodal property methods, time map analysis, and Conley index at infinity. I will show how we may combine these methods in order to constrcut both the bounded portion of the non-compact global attractor as well as the attractor at infinity. I will also discuss the ways in which asymptotic asymmetry adds significant complexity to the connection problem both at infinity and in the transition between the bounded attractor and the attractor at infinity. This talk is based on joint work with Kristen Moore (U. Michigan) and Juliette Hell (F.U. Berlin). - February 25 Dan Alexander (Drake University)

Title: A History of Complex Dynamics: From Schr\"oder to Siegal - March 4 Dan Thompson (Ohio State)

Title: Large Deviations for S-gap shifts

Abstract: We establish a Large Deviations Principle (LDP) for a large class of symbolic systems previously studied by Climenhaga and myself. The techniques provide a new framework to establish LDP for dynamical systems with non-uniform structure. I will begin with a brief summary of the theory of Large Deviations in dynamical systems. I will explain the results via the application to the S-gap shifts: a natural family of symbolic spaces which are easy to define but which can be challenging to study! This is joint work with Vaughn Climenhaga (Houston) and Kenichiro Yamamoto (Tokyo Denki). - March 18 Gareth Roberts
(Holy Cross)

Title: Stability of Relative Equilibria in the Planar N-Vortex Problem

Abstract: In the weather research and forecasting models of certain hurricanes, vortex crystals are found within a polygonal-shaped eyewall. These special configurations can be interpreted as relative equilibria (rigidly rotating solutions) of the point vortex problem introduced by Helmholtz. Their stability is thus of considerable importance. Adapting Moeckel's approach for the companion problem in celestial mechanics, we present some theory and results on the linear and nonlinear stability of relative equilibria in the planar N-vortex problem. A topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case. - March 25 Jorge
Freitas (Porto)

Title: Statistical behaviour of chaotic dynamics

Abstract: We will consider mixing dynamical systems with suitable rates of decay of correlations, which measure how fast the system looses memory. We sill see that such systems satisfy certain statistical properties such as: Strong Law of Large Numbers (asserting that time averages converge a.s. to spacial averages), Central Limit Theorem (which gives a distributional limit for adequately normalised time averages), Large Deviations Rates (which have to do with the speed of convergence of time averages), Extreme Value Laws (which give distributional limits for the partial maxima of observations), Hitting Time Statistics and Return Time Statistics (which give distributional limits for the recurrence time to shrinking targets). - April 1 Mónica Moreno Rocha
(CIMAT, Guanajuato, México)

Title: In the hunt for Herman rings

Abstract: Consider an analytic map over an annular domain U of the complex plane. U is called an Arnold-Herman (or simply a Herman) ring if the iterates of the map restricted to U are analytically conjugate to an irrational rotation acting on an annulus. In general, it is a difficult problem to determine the existence of Herman rings since, in contrast to other type of Fatou domains, a Herman ring is not associated to a periodic orbit. In this talk I will provide an overview on the theory of rotation domains and the existence of Herman rings for rational and meromorphic maps. Then I will concentrate on the iteration of even elliptic maps and show how the symmetries inherit from the lattice prevent these maps from having Herman rings. This is a joint work with Pablo Perez Lucas (CIMAT). - April 8 Arnd Scheel (Minnesota)

Title: Pattern selection in the wake of fronts

Abstract: Motivated by the formation of complex patterns in the wake of growth processes in bacterial colonies, we'll study more generally patterns formed in the wake of moving fronts. We discuss a number of mathematical problems that arise when one tries to predict which wavenumbers and what type of pattern is created by the growth process. We therefore describe in some detail how to derive linear predictions and analyze nonlinear mechanisms that lead to failure of these predictions. This is joint work with Matt Holzer and Ryan Goh. - April 18 David
Fried (BU) ** SPECIAL THURSDAY SEMINAR; tea and talk are in MCS 137**

Title: Hyperbolic Billiards and Symbolic Dynamics

Abstract: A polygon T in the hyperbolic plane supports continuous billiard flow when each angle is of the form pi/n for some integrer n > 1. We will give symbolic dynamics for such flows. Our methods generalize those of Jonnie Naylor, who treated the case when all interior angles are pi/2. These pethods apply to the geodesic flow over many, perhaps all, compact quotients of the hyperbolic plane. - April 22 Geordie Richards (Institute for Mathematics and its
Applications, Minnesota)

Title: Invariance of the Gibbs measure for the periodic quartic gKdV

Abstract: The periodic generalized Korteweg-de Vries equation (gKdV) can be interpreted as an infinite-dimensional Hamiltonian system. Some properties of finite-dimensional Hamiltonian dynamics can be extended to infinite dimensions; for example, the invariance of the Gibbs measure under the flow. We present the invariance of the Gibbs measure for the gauge-transformed quartic gKdV. As a corollary, we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed. - April 29 Freddy
Dumortier (Hasselt University)

Title: Relaxation oscillations in slow-fast systems

Abstract: The talk deals with two-dimensional slow-fast systems. These systems depend on a small parameter $\varepsilon$, and possibly also on other parameters, in a way that for $\varepsilon=0$ the equation has a continuum of singular points. Such systems can be studied by means of Geometric Singular Perturbation Theory. This theory essentially relies on center manifold reduction. The first to introduce it was Fenichel. Traditional Fenichel theory can however only be used near normally hyperbolic situations. Around 1995, it became clear how the blow technique could extend the power of the theory to include contact points. Center manifolds, normal forms and blow-up permit one to treat singular perturbation problems by means of traditional methods from dynamical systems theory. In the talk, we will briefly recall the essential ingredients from the theory. We will mainly present a number of recent results concerning relaxation oscillations for $\varepsilon>0$, $\varepsilon$ small: their number and the bifurcations they undergo. We will focus on the current state of the theory paying attention to remaining problems. The results come from a number of joint papers with Robert Roussarie and Peter De Maesschalck, including work in progress.