## Seminars and Colloquia by Series

### Rigidity and Cutting and Stacking Constructions

Series
CDSNS Colloquium
Time
Wednesday, December 6, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Kelly YanceyInstitute for Defense Analyses
A special class of dynamical systems that we will focus on are substitutions. This class of systems provides a variety of ergodic theoretic behavior and is connected to self-similar interval exchange transformations. During this talk we will explore rigidity sequences for these systems. A sequence $\left( n_m \right)$ is a rigidity sequence for the dynamical system $(X,T,\mu)$ if $\mu(T^{n_m}A\cap A)\rightarrow \mu(A)$ for all positive measure sets $A$. We will discuss the structure of rigidity sequences for substitutions that are rank-one and substitutions that have constant length. This is joint work with Jon Fickenscher.

### Bistable gaits and wobbling induced by pedestrian-bridge interactions

Series
CDSNS Colloquium
Time
Monday, November 20, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Igor BelykhGeorgia State University
Several modern footbridges around the world have experienced large lateral vibrations during crowd loading events. The onset of large-amplitude bridge wobbling has generally been attributed to crowd synchrony; although, its role in the initiation of wobbling has been challenged. In this talk, we will discuss (i) the contribution of a single pedestrian into overall, possibly unsynchronized, crowd dynamics, and (ii) detailed, yet analytically tractable, models of crowd phase-locking. The pedestrian models can be used as "crash test dummies" when numerically probing a specific bridge design. This is particularly important because the U.S. code for designing pedestrian bridges does not contain explicit guidelines that account for the collective pedestrian behavior. This talk is based on two recent papers: Belykh et al., Science Advances, 3, e1701512 (2017) and Belykh et al., Chaos, 26, 116314 (2016).

### Irregularity of the solutions and Noncompactness of the Global Attracting Set in a Coupled ODE-PDE Model of the Neocortex

Series
CDSNS Colloquium
Time
Monday, November 6, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Farshad ShiraniGeorgia Institute of Technology
We present a mean field model of electroencephalographic activity in the brain, which is composed of a system of coupled ODEs and PDEs. We show the existence and uniqueness of weak and strong solutions of this model and investigate the regularity of the solutions. We establish biophysically plausible semidynamical system frameworks and show that the semigroups of weak and strong solution operators possess bounded absorbing sets. We show that there exist parameter values for which the semidynamical systems do not possess a global attractor due to the lack of the compactness property. In this case, the internal dynamics of the ODE components of the solutions can create asymptotic spatial discontinuities in the solutions, regardless of the smoothness of the initial values and forcing terms.

### Recurrence on abelian cover. Closed geodesics in manifolds of negative curvature

Series
CDSNS Colloquium
Time
Monday, October 23, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Albert FathiGeorgia Institute of Technology
If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover is also chain-recurrent. This has consequences on closed geodesics in manifold of negative curvature.

### Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation

Series
CDSNS Colloquium
Time
Monday, September 25, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Larissa SerdukovaGeorgia Institute of Technology
For the tipping elements in the Earth’s climate system, the most important issue to address is how stable is the desirable state against random perturbations. Extreme biotic and climatic events pose severe hazards to tropical rainforests. Their local effects are extremely stochastic and difficult to measure. Moreover, the direction and intensity of the response of forest trees to such perturbations are unknown, especially given the lack of efficient dynamical vegetation models to evaluate forest tree cover changes over time. In this study, we consider randomness in the mathematical modelling of forest trees by incorporating uncertainty through a stochastic differential equation. According to field-based evidence, the interactions between fires and droughts are a more direct mechanism that may describe sudden forest degradation in the south-eastern Amazon. In modeling the Amazonian vegetation system, we include symmetric α-stable Lévy perturbations. We report results of stability analysis of the metastable fertile forest state. We conclude that even a very slight threat to the forest state stability represents L´evy noise with large jumps of low intensity, that can be interpreted as a fire occurring in a non-drought year. During years of severe drought, high-intensity fires significantly accelerate the transition between a forest and savanna state.

### Periodic Driving at High Frequencies of an Impurity in the Isotropic XY Chain

Series
CDSNS Colloquium
Time
Monday, September 18, 2017 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Livia CorsiGeorgia Institute of Technology
I will consider the isotropic XY quantum chain with a transverse magnetic field acting on a single site and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. It has been shown in the early 70’s that, in the thermodynamic limit, the state of such system obeys a linear time-dependent Schrodinger equation with a memory term. I will consider two different regimes, namely when the perturbation has non-zero or zero average, and I will show that if the magnitute of the potential is small enough then for large enough frequencies the state approaches a periodic orbit synchronized with the potential. Moreover I will provide the explicit rate of convergence to the asymptotics. This is a joint work with G. Genovese.

### Invariant manifolds in energy harvesting coupled oscillators

Series
CDSNS Colloquium
Time
Tuesday, July 25, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Albert GranadosDepartment of Applied Mathematics and Computer Science, Technical University of Denmark
When perturbed with a small periodic forcing, two (or more) coupledconservative oscillators can exhibit instabilities: trajectories thatbecome unstable while accumulating unbounded'' energy from thesource. This is known as Arnold diffusion, and has been traditionallyapplied to celestial mechanics, for example to study the stability ofthe solar system or to explain the Kirkwood gaps in the asteroid belt.However, such phenomenon could be extremely useful in energyharvesting systems as well, whose aim is precisely to capture as muchenergy as possible from a source.In this talk we will show a first step towards the application ofArnold diffusion theory in energy harvesting systems. We will consideran energy harvesting system based on two piezoelectric oscillators.When forced to oscillate, for instance when driven by a small periodicvibration, such oscillators create an electrical current which chargesan accumulator (a capacitor or a battery). Unfortunately, suchoscillators are not conservative, as they are not perfectly elastic(they exhibit damping).We will discuss the persistence of normally hyperbolic invariantmanifolds, which play a crucial role in the diffusing mechanisms. Bymeans of the parameterization method, we will compute such manifoldsand their associated stable and unstable manifolds. We will alsodiscuss the Melnikov method to obtain sufficient conditions for theexistence of homoclinic intersections.

### A fractalization process for affine skew-products on the complex plane

Series
CDSNS Colloquium
Time
Wednesday, May 10, 2017 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marc Jorba-CuscoUniversitat de Barcelona
Consider an affine skew product of the complex plane. $$\begin{cases} \omega \mapsto \theta+\omega,\\ z \mapsto =a(\theta \mu)z+c, \end{cases}$$where $\theta \in \mathbb{T}$, $z\in \mathbb{C}$, $\omega$ is Diophantine, and $\mu$ and $c$ are real parameters. In this talk we show that, under suitable conditions, the affine skew product has an invariant curve that undergoes a fractalization process when $\mu$ goes to a critical value. The main hypothesis needed is the lack of reducibility of the system. A characterization of reducibility of linear skew-products on the complex plane is provided. We also include a linear and topological classification of these systems. Join work with: N\'uria Fagella, \`Angel Jorba and Joan Carles Tatjer

### Weak KAM theorem for Frenkel-Kontorova models and related topics

Series
CDSNS Colloquium
Time
Monday, May 8, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xifeng SuBeijing Normal University
We will consider the Frenkel-Kontorova models and their higher dimensional generalizations and talk about the corresponding discrete weak KAM theory. The existence of the discrete weak KAM solutions is related to the additive eigenvalue problem in ergodic optimization. In particular, I will show that the discrete weak KAM solutions converge to the weak KAM solutions of the autonomous Tonelli Hamilton-Jacobi equations as the time step goes to zero.

### Lyapunov exponents and spectral analysis of ergodic Schrödinger operators

Series
CDSNS Colloquium
Time
Monday, April 24, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zehnghe ZhangRice University
One dimensional discrete Schrödinger operators arise naturally in modeling the motion of quantum particles in a disordered medium. The medium is described by potentials which may naturally be generated by certain ergodic dynamics. We will begin with two classic models where the potentials are periodic sequences and i.i.d. random variables (Anderson Model). Then we will move on to quasi-periodic potentials, of which the randomness is between periodic and i.i.d models and the phenomena may become more subtle, e.g. a metal-insulator type of transition may occur. We will show how the dynamical object, the Lyapunov exponent, plays a key role in the spectral analysis of these types of operators.