Seminars and Colloquia by Series

Floquet bundles for tridiagonal competitive-cooperative systems

Series
CDSNS Colloquium
Time
Monday, October 8, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yi Wang – University of Sciences and Technology of China
For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.

Greene's Criterion for the Breakup of Invariant Tori of Volume Preserving Maps

Series
CDSNS Colloquium
Time
Monday, October 1, 2012 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Adam Fox – Univ. of Colorado
Invariant tori play a prominent role in the dynamics of symplectic maps. These tori are especially important in two dimensional systems where they form a boundary to transport. Volume preserving maps also admit families of invariant rotational tori, which will restrict transport in a d dimensional map with one action and d-1 angles. These maps most commonly arise in the study of incompressible fluid flows, however can also be used to model magnetic field-line flows, granular mixing, and the perturbed motion of comets in near-parabolic orbits. Although a wealth of theory has been developed describing tori in symplectic maps, little of this theory extends to the volume preserving case. In this talk we will explore the invariant tori of a 3 dimensional quadratic, volume preserving map with one action and two angles. A method will be presented for determining when an invariant torus with a given frequency is destroyed under perturbation, based on the stability of approximating periodic orbits.

Modeling transcriptional elongation

Series
CDSNS Colloquium
Time
Monday, September 24, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tomas Gedeon – Montana State University –

Please Note: Joint with Applied and Computational Mathematics Seminar

Bio-polymerization processes like transcription and translation are central to a proper function of a cell. The speed at which the bio-polymer grows is affected both by number of pauses of elongation machinery, as well their numbers due to crowding effects. In order to quantify these effects in fast transcribing ribosome genes, we rigorously show that a classical traffic flow model is a limit of mean occupancy ODE model. We compare the simulation of this model to a stochastic model and evaluate the combined effect of the polymerase density and the existence of pauses on transcription rate of ribosomal genes.

Toward Computer Assisted Morse Theory.

Series
CDSNS Colloquium
Time
Friday, September 7, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jason Mireles-James – Rutgers University
I'll discuss some work on rigorous computation of invariant manifolds and computer assisted proof of the existence of transverse connecting orbits for differential equations. I'm also interested in how these computations can be used to obtain global topological data, such as the chain groups and boundary maps of Morse Theory.

Computer assisted proofs for normally hyperbolic invariant manifolds

Series
CDSNS Colloquium
Time
Monday, August 27, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
M. Capinski – AGH Univ. Krakow and SOM, Gatech
We shall present a method which establishes existence of normally hyperbolic invariant manifolds for maps within a specified domain. The method can be applied in a non-perturbative setting. The required conditions follow from bounds on the first derivative of the map, and are verifiable using rigorous numerics. We show how the method can be applied for a driven logistic map, and also present examples of proofs of invariant manifolds in the restricted three body problem.

Schroedinger operators beyond the Almost Mathieu

Series
CDSNS Colloquium
Time
Wednesday, July 11, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Haro – Univ. of Barcelona
This talk is devoted to quasi-periodic Schr\"odinger operators beyond the Almost Mathieu, with more general potentials and interactions, considering the connections between the spectral properties of these operators and the dynamical properties of the asso- ciated quasi-periodic linear skew-products. In par- ticular, we present a Thouless formula and some consequences of Aubry duality. We illustrate the results with numerical computations. This is a join work with Joaquim Puig

On a weak form of Arnold diffusion in arbitrary degrees of freedom

Series
CDSNS Colloquium
Time
Monday, April 23, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vadim Kaloshin – Univ. of Maryland
Consider a generic perturbation of a nearly integrable system of {\it arbitrary degrees of freedom $n\ge 2$ system}\[H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T,\]with strictly convex $H_0$. Jointly with P.Bernard and K.Zhang we prove existence of orbits $(\th,p)(t)$ exhibiting Arnold diffusion \[\|p(t)-p(0) \| >l(H_1)>0 \quad \textup{independently of }\eps.\]Action increment is independent of size of perturbation$\eps$, but does depend on a perturbation $\eps H_1$.This establishes a weak form of Arnold diffusion. The main difficulty in getting rid of $l(H_1)$ is presence of strong double resonances. In this case for $n=2$we prove existence of normally hyperbolic invariant manifolds passing through these double resonances. (joint with P. Bernard and K. Zhang)

Stability of ODE with colored noise forcing.

Series
CDSNS Colloquium
Time
Friday, April 20, 2012 - 11:10 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Timothy Blass – Carnegie Melon University
I will discuss recent work on the stability of linear equations under parametric forcing by colored noise. The noises considered are built from Ornstein-Uhlenbeck vector processes. Stability of the solutions is determined by the boundedness of their second moments. Our approach uses the Fokker-Planck equation and the associated PDE for the marginal moments to determine the growth rate of the moments. This leads to an eigenvalue problem, which is solved using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes into "ladder operators." The results are given in terms of a perturbation expansion in the size of the noise. We have found very good agreement between our results and numerical simulations. This is joint work with L.A. Romero.

Circle homeomorphisms with singularity points.

Series
CDSNS Colloquium
Time
Monday, April 16, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akhtam Djalilov – Univ. of Samarkand and CUNY Stony Brook
An important question in circle dynamics is regarding the absolute continuity of an invariant measure. We will consider orientation preserving circle homeomorphisms with break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is well known that the invariant measures of sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesgue measure. But in the case of homeomorphisms with break points the results are quite dierent. We will discuss conjugacies between two circle homeomorphisms with break points. Consider the class of circle homeomorphisms with one break point b and satisfying the Katznelson-Ornsteins smoothness condition i.e. Df is absolutely continuous on [b; b + 1] and D2f 2 Lp(S1; dl); p > 1: We will formulate some results concerning the renormaliza- tion behavior of such circle maps.

Measures of maximal entropy and integrated density of states for the discrete Schrodinger operator with Fibonacci potential

Series
CDSNS Colloquium
Time
Monday, April 9, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Anton Gorodetski – UC Irvine –
The discrete Schrodinger operator with Fibonacci potential is a central model in the study of electronic properties of one-dimensional quasicrystals. Certain renormalization procedure allows to reduce many questions on specral properties of this operator to the questions on dynamical properties of a so called trace map. It turnes out that the trace map is hyperbolic, and its measure of maximal entropy is directly related to the integrated density of states of the Fibonacci Hamiltonian. In particular, this provides the first example of an ergodic family of Schrodinger operators with singular density of states measure for which exact dimensionality can be shown. This is a joint work with D. Damanik.

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