Dynamical Systems Seminars

Spring 2015

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.

- **** CANCELLED DUE TO SNOW ****

February 9: Blake Barker (Brown)

Title: Toward rigorous verification of the stability of traveling waves

Abstract: We discuss various aspects of numerical stability analysis of traveling wave solutions with a particular eye toward the development of guaranteed error bounds. In particular, we consider the stability of periodic traveling waves of the generalized Kuramoto-Sivashinsky equation in the limit the diffusion coefficient goes to zero, or the Korteweg-de Vries limit. For a large class of physically relevant problems, nonlinear stability of traveling waves is determined by spectral stability. Often, the spectral stability problem must be studied numerically because current analytical approaches do not suffice. Using error bounds and interval arithmetic provides a way to rigorously verify spectral stability numerically. Such numerical verification often leads to problem dependent challenges. We describe our success and ongoing efforts at carrying out this type of numerical analysis.

**** CANCELLED DUE TO SNOW **** - February 23: Oliver
Knill (Harvard)

Title: Birkhoff sums over the golden rotation

Abstract: We look at Birkhoff sums S_n(t)/n=sum_{k=1}^n X_k(t)/n with X_k(t)=g(T^kt) where T is the irrational golden rotation and where g(t)=cot(pi t). Similar sums have been studied by number theorists like Hardy and Littlewood or Sinai and Ulcigrain in the context of the curlicue problem. Birkhoff sums can be visualized if the time interval [0,n] is rescaled so that it displays a graph over the interval [0,1]. While for any L^1 function g(t) and ergodic T, the sum S_[n x](t)/n converges a.e. to a linear function Mx by Birkhoff's ergodic theorem, there is an interesting phenomenon for Cauchy distributed random variables like if g(t) = cot(pi t) The function x -> S_[n x]/n on [0,1] converges for n -> infinity to an explicitly given fractal limiting function, if n is restricted to Fibonacci numbers F(2n) and if the start point t is 0. The convergence to the ``golden graph" shows a truly self-similar random walk. It explains some observations obtained together with John Lesieutre and Folkert Tangerman, where we summed the anti derivative G of g which is the Hilbert transform of a piecewise linear periodic function. Birkhoff sums are relevant in KAM contexts, both in analytic and smooth situations or in Denjoy-Koksma theory which is a refinement of Birkhoff for Diophantine irrational rotations. In a probabilistic context, we have a discrete time stochastic process modeling ``high risk" situations as hitting a point near the singularity catastrophically changes the sum. Diophantine conditions assure that there is enough time to ``recover" from such a catastrophe. There are other connections like with modular functions in number theory or Milnor's theorem telling that the cot function is the unique non constant solution to the Kubert relation (1/n) sum_{k=1}^n g(t+k/n) = g(t). - March 2: Yuri Latushkin
(Missouri)

Title: The Morse and Maslov indices for multidimensional differential operators.

Abstract: In this talk we will discuss some recent results on connections between the Maslov and the Morse indices for differential operators. The Morse index is a spectral quantity defined as the number of unstable eigenvalues counting multiplicities while the Maslov index is a geometric characteristic defined as the signed number of intersections of a path in the space of Lagrangian planes with the train of a given plane. The problem of relating these two quantities is rooted in Sturm's Theory and has a long history going back to the classical work by Arnold, Bott, Duistermaat and Smale. Two situations will be addressed: First, the case when the differential operator is a Schroedinger operator equipped with theta-periodic boundary conditions, and second, when the Schroedinger operators are acting on a family of multidimensional domains obtained by shrinking a star-shaped domain to a point and are equipped with either Dirichlet or quite general Robbin boundary conditions. This is a joint work with G. Cox, C. Jones, R. Marangell, A. Sukhtayev, and S. Sukhtaiev. - March 16: Panayotis G. Kevrekidis (UMass, Amherst)

Title: "Nonlinear Waves in Granular Crystals: Theory, Computation, Experiments and Applications"

Abstract: In this talk, we will provide an overview of results in the setting of granular crystals, consisting of beads interacting through Hertzian contacts. We will start from the simplest setting of one-dimensional, monoatomic chains where highly localized traveling waves exist and we will also examine states in the form of (dark) discrete breathers and shock waves therein. Wherever possible, we will corroborate these considerations with recent experimental results. We will then extend our considerations to the case of diatomic chains and examine how the properties of traveling waves and also of discrete breathers are modified in the latter setting. More highly heterogeneous chains will be briefly examined as well. In addition to considering the purely Hamiltonian case, select examples of the damped-driven variant of the system and its rich phenomenology, including chaotic response and bistability/hysteresis will also be shown.

In the last part of the talk, time-permitting, a number of recent aspects will be touched upon including: (a) formation of traveling waves with non-vanishing tails in elastic woodpile periodic structures; (b) super-diffusive transport in disordered granular chains; (c) applications of these lattices for the demonstration of switching and acoustic logic gates; (d) prototypical examples of extensions to two dimensions in hexagonal, as well as square arrays. - March 23: Marta Canadell
(Georgia Tech)

Title: A KAM-like theorem for normally hyperbolic quasi-periodic tori leading to efficient algorithms

Abstract: We present a KAM-like theorem in a-posteriori format for the existence of quasi-periodic normally hyperbolic invariant tori (QP-NHIT), with a fixed frequency, in families of discrete dynamical systems. This theorem gives us rigorous support for numerical algorithms of continuation of QP-NHIT, and so, provides us with an efficient algorithm to compute them by adjusting parameters of the family. We also present implementations of this method to continue the invariant manifold and its bundles with respect to parameters, and to explore different mechanisms of breakdown. This is a joint work with Alex Haro.

- March 30: David Lipshutz
(Brown) *** Special time: 3pm, in MCS 148, with tea at 2:45pm in MCS 153 ***

Title: Oscillatory behavior of dynamical systems with delayed dynamics, non-negativity constraints and small noise

Abstract: Dynamical system models with delayed feedback, non-negativity constraints and small noise arise in a variety of applications in science and engineering. In some applications, oscillatory behavior is critical to the well functioning of the system. As a prototype for such models, we consider a one-dimensional stochastic delay differential equation with a non-negativity constraint and a related deterministic equation. We study the existence, uniqueness and stability of ``slowly oscillating'' periodic solutions to the deterministic equation and show that solutions to the stochastic equation remain close to these periodic solutions from the perspective of large deviations. We illustrate our finding with an example of a simple genetic circuit model. This is joint work with Ruth Williams. - April 6: Björn de
Rijk (Leiden)

Title: Fine structure of the boundary of the Busse balloon beyond Gierer-Meinhardt

Abstract: The Busse balloon is the region in (wavenumber, parameter)-space for which stable spatially periodic patterns exist. It has been observed analytically in the Gierer-Meinhardt equation that the boundary of the Busse balloon in the small wavenumber limit is non-smooth and has a fine-structure of intertwining curves that represent two kinds of Hopf bifurcations. Moreover, the Busse Balloon has a 'homoclinic tip' at which the last periodic patterns to become unstable have wave number k = 0. Are these phenomena reserved for Gierer-Meinhardt type of models only? To answer this question we have extended the existing stability theory significantly to a general class of singularly perturbed reaction-diffusion systems. In this talk I will explain how this novel theory can be applied to find out whether these phenomena persist.

- April 13: Scott Sutherland (Stony Brook)
- April 27: John Gemmer (Brown)