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.
- **** CANCELLED ****
April 13: Scott Sutherland (Stony Brook)
Title: On the measure of the Feigenbaum Julia set
Abstract: We discuss a computer-assisted proof that the Julia set of the limit
of period-doubling renormalization in the quadratic family has zero
measure. This is joint work with Artem Dudko.
- April 27: John Gemmer (Brown)
Title: Isometric immersions and pattern formation in swelling thin elastic sheets
Abstract: I will discuss joint work with Shankar Venkataramani, Eran Sharon
and Govind Menon concerning pattern selection in growing thin elastic sheets
and grain boundary networks. Many pattern forming systems in material science
can be modeled as the gradient flow of a free energy functional that is the
sum of a strong non-convex energy and a singular perturbation that is convex
in higher order derivatives. Examples of such systems include domain wall
structures in ferromagnets, austenite-martensite phase transitions, and
dislocation patterns in liquid crystal elastomers. Swelling thin elastic
sheets are archetypical examples of such systems and are often modeled as the
minimum an elastic energy which is the sum of a strong stretching energy
penalizing deviations from a growth induced geometry and a weak bending
energy. A fundamental question is whether we can deduce the three dimensional
configuration of the sheet given exact knowledge of the swelling
pattern. Using a combination of analysis and numerical conjectures I will
argue that the periodic patterns observed in nature correspond to low energy
isometric immersions of an imposed geometry.
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