Seminars and Colloquia by Series

Discrete stochastic Hamilton Jacobi equation

Series
CDSNS Colloquium
Time
Monday, February 5, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Renato Iturriaga – CIMAT
We present a discrete setting for the viscous Hamilton Jacobi equation, and prove convergence to the continuous case.

KAM for quasi-linear and fully nonlinear PDEs

Series
CDSNS Colloquium
Time
Monday, February 5, 2018 - 10:10 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Riccardo Montalto – University of Zurich
In this talk I will present some results concerning the existence and the stability of quasi-periodic solutions for quasi-linear and fully nonlinear PDEs. In particular, I will focus on the Water waves equation. The proof is based on a Nash-moser iterative scheme and on the reduction to constant coefficients of the linearized PDE at any approximate solution. Due to the non-local nature of the water waves equation, such a reduction procedure is achieved by using techniques from Harmonic Analysis and microlocal analysis, like Fourier integral operators and Pseudo differential operators.

Multiplicity results for some classes of non-local elliptic equations

Series
CDSNS Colloquium
Time
Monday, January 29, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Xifeng Su – Beijing Normal University
We will consider the nonlinear elliptic PDEs driven by the fractional Laplacian with superlinear or asymptotically linear terms or combined nonlinearities. An L^infinity regularity result is given using the De Giorgi-Stampacchia iteration method. By the Mountain Pass Theorem and other nonlinear analysis methods, the local and global existence and multiplicity of non-trivial solutions for these equations are established. This is joint work with Yuanhong Wei.

A posteriori KAM theorems for systems with first integrals in involution

Series
CDSNS Colloquium
Time
Monday, January 22, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Alex Haro – University of Barcelona
Some relevant Hamiltonian systems in Celestial Mechanics have first integrals in involution. A classic technique to study such systems, known as symplectic reduction, is based in reducing the number of degrees of freedom by using the first integrals. In this talk we present two a posteriori KAM theorems for Hamiltonian systems with first integrals in involution, including the isoenergetic case, without using symplectic reduction. The approach leads to efficient numerical methods and validating techniques.This is a joint work with Alejandro Luque.

Symbolic computations of homoclinic chaos

Series
CDSNS Colloquium
Time
Monday, December 11, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andrey Shilnikov – Georgia State University
Over recent years, a great deal of analytical studies and modeling simulations have been brought together to identify the key signatures that allow dynamically similar nonlinear systems from diverse origins to be united into a single class. Among these key structures are bifurcations of homoclinic and heteroclinic connections of saddle equilibria and periodic orbits. Such homoclinic structures are the primary cause for high sensitivity and instability of deterministic chaos in various systems. Development of effective, intelligent and yet simple algorithms and tools is an imperative task for studies of complex dynamics in generic nonlinear systems. The core of our approach is the reduction of the time evolution of a characteristic observable in a system to its symbolic representation to conjugate or differentiate between similar behaviors. Of our particular consideration are the Lorenz-like systems and systems with spiral chaos due to the Shilnikov saddle-focus. The proposed approach and tools will let one detect homoclinic and heteroclinic orbits, and carry out state of the art studies homoclinic bifurcations in parameterized systems of diverse origins.

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 Yancey – Institute 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 Belykh – Georgia 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 Shirani – Georgia 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 Fathi – Georgia 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 Serdukova – Georgia 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.

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