Seminars and Colloquia by Series

A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries

Series
CDSNS Colloquium
Time
Friday, March 10, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Emmanuel FleurantinUNC, GMU

https://gatech.zoom.us/j/98358157136 

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River.  Using the Maslov index we identify a subset of the River which is comprised of local minimizers.  The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the talk is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

Exploring global dynamics and blowup in some nonlinear PDEs

Series
CDSNS Colloquium
Time
Friday, February 24, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Jonathan JaquetteBrown University

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

Some results on a simple model of kinetic theory

Series
CDSNS Colloquium
Time
Friday, February 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006; Zoom streaming available
Speaker
Federico BonettoGeorgia Tech

Please Note: Zoom link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified in such a way to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, we worked with M. Loss and others to extend the analysis to more "realistic" versions of the original model.

I will introduce the Kac model and present some standard and more recent results. These results refer to a system with a fixed number of particles and at fixed kinetic energy (micro canonical ensemble) or temperature (canonical ensemble). I will introduce a "Grand Canonical" version of the Kac system and discuss new results on it.

The controllability function method and the feedback synthesis problem for a robust linear system

Series
CDSNS Colloquium
Time
Friday, February 10, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Tetiana RevinaV. N. KARAZIN KHARKIV NATIONAL UNIVERSITY

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

The talk is about controllability for uncertain linear systems. Our approach is 
based on the Controllability Function (CF) method proposed by V.I. Korobov in 
1979. The CF method is a development of the Lyapunov function method and the 
dynamic programming method. The CF includes both approaches at a certain values 
of parameters. The main advance of the CF method is finiteness of the time of motion 
(settling-time function). 
In the talk the feedback synthesis problem for a chain of integrators system 
with continuous bounded unknown perturbations is considered. This problem consist 
in constructing a control in explicit form which depends on phase coordinates and 
steers an arbitrary initial point from a neighborhood of the origin to the origin in a 
finite time (settling-time function). Besides the control is satisfies some preassigned 
constrains. The range of the unknown perturbations such that the control solving the 
synthesis problem for the system without the perturbations also solves the synthesis 
problem for the perturbed system are found. This study shows the relations between 
the range of perturbations and the bounds of the settling-time function.
In particular the feedback synthesis problem for the motion of a material 
point with allowance for friction is solved. 
Keywords: chain of integrators, finite-time stability, robust control, settling 
time estimation, uncertain systems, unknown bounded perturbation

Nonsingular Poisson suspensions

Series
CDSNS Colloquium
Time
Friday, February 3, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Oleksandr DanilenkoInstitute for Low Temperature Physics and Engineering

 https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Let T be an invertible measure preserving transformation of a standard infinite measure space (X,m). Then a Poisson suspension (X*,m*,T*) of the dynamical system (X,m,T) is a well studied object in ergodic theory (especially for the last 20 years). It has physical applications as a model for the ideal gas consisting of countably many non-interacting particles. A natural problem is to develop a nonsingular counterpart of the theory of Poisson suspensions. The following will be enlightened in the talk:

--- description of the m-nonsingular (i.e. preserving the equivalence class of m) transformations T such that T* is m*-nonsingular
---algebraic and topological properties of the group of all m*-nonsingular Poisson suspensions
--- an interplay between dynamical properties of T and T*
--- an example of a "phase transition" in the ergodic properties of T* depending on the scaling of m
--- applications to Kazhdan property (T), stationary (nonsingular) group actions and the Furstenberg entropy.

(joint work with Z. Kosloff and E. Roy)

 

Differential encoding of sensory information across cortical microcircuitry

Series
CDSNS Colloquium
Time
Friday, January 27, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Hannah ChoiGeorgia Tech

Please Note: https://gatech.zoom.us/j/98358157136

Mammalian cortical networks are known to process sensory information utilizing feedforward and feedback connections along the cortical hierarchy as well as intra-areal connections between different cortical layers. While feedback and feedforward signals have distinct layer-specific connectivity motifs preserved across species, the computational relevance of these connections is not known. Motivated by predictive coding theory, we study how expected and unexpected visual information is encoded along the cortical hierarchy, and how layer-specific feedforward and feedback connectivity contributes to differential, context-dependent representations of information across cortical layers, by analyzing experimental recordings of neural populations and also by building a recurrent neural network (RNN) model of the cortical microcircuitry. Experimental evidence shows that information about identity of the visual inputs and expectations are encoded in different areas of the mouse visual cortex, and simulations with our RNNs which incorporate biologically plausible connectivity motifs suggest that layer-specific feedforward and feedback connections may be the key contributor to this differential representation of information.
 

Statistical and non-statistical dynamics in doubly intermittent maps

Series
CDSNS Colloquium
Time
Friday, November 11, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 005
Speaker
Stefano LuzzattoAbdus Salam International Centre for Theoretical Physics (ICTP)

Please Note: In-person. Streaming available via zoom: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

We introduce a large family of one-dimensional full branch maps which generalize the classical “intermittency maps” by admitting two neutral fixed points and possibly also critical points and/or singularities. We study the statistical properties of these maps for various parameter values, including the existence (and non-existence) of physical measures, and their properties such as decay of correlations and limit theorems. In particular we describe a new mechanism for relatively persistent non-statistical chaotic dynamics. This is joint work with Douglas Coates and Muhammad Mubarak.

Absolutely Periodic Billiard Orbits of Arbitrarily High Order

Series
CDSNS Colloquium
Time
Friday, November 4, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person talk in Skiles 005; streaming available via Zoom
Speaker
Keagan CallisUniversity of Maryland

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is also an indication that Ivrii's Conjecture about the measure set of periodic orbits may not be true.

Spontaneous periodic orbits in the Navier-Stokes flow

Series
CDSNS Colloquium
Time
Friday, October 21, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Online via Zoom (Skiles 006 viewing party)
Speaker
Maxime BrendenEcole Polytechnique

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

In this talk, I will present results concerning the existence and the precise description of periodic solutions of the Navier-Stokes equations with a time- independent forcing, obtained in collaboration with Jan Bouwe van den Berg (VU Amsterdam), Jean-Philippe Lessard (McGill) and Lennaert van Veen (Ontario TU).

These results are obtained by combining numerical simulations, a posteriori error estimates, interval arithmetic, and a fixed point theorem applied to a quasi-Newton operator, which yields the existence of an exact solution in a small and explicit neighborhood of the numerical one.

I will first introduce the main ideas and techniques required for this type of approach on a simple example, and then discuss their usage in more complex settings like the Navier-Stokes equations.

Functional Poisson approximations for some dissipative systems

Series
CDSNS Colloquium
Time
Friday, September 30, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 006
Speaker
Yaofeng SuGeorgia Tech

The study of Poisson approximations of the process of recurrences to small subsets in the phase spaces of chaotic dynamical systems, started in 1991, have developed by now into a large active area of the dynamical systems theory. In this talk, I will present some new results. This is a joint work with Prof. Leonid Bunimovich.

  1. I will start with some examples of dissipative hyperbolic systems,
  2. then formulate the question of functional Poisson approximations for these systems.
  3. To study Poisson approximations, I will present two difficulties, called short returns and ring conditions.
  4. These two difficulties can be partially solved under some conditions of, e.g. the dimension of the dynamics, the Hausdorff dimension of the SRB measure, etc. I will present a new method which does not depend on dimensions but can completely solve these two difficulties for dissipative systems.

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