Seminars and Colloquia by Series

Phase-shifted, exponentially small nanopterons in a model of KdV coupled to an oscillatory field

Series
CDSNS Colloquium
Time
Friday, September 22, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Tim FaverKennesaw State University

We develop nanopteron solutions for a coupled system of singularly perturbed ordinary differential equations.  To leading order, one equation governs the traveling wave profile for the Korteweg-de Vries (KdV) equation, while the other models a simple harmonic oscillator whose small mass is the problem’s natural small parameter.  A nanopteron solution consists of the superposition of an exponentially localized term and a small-amplitude periodic term.  We construct two families of nanopterons.  In the first, the periodic amplitude is fixed to be exponentially small but nonzero, and an auxiliary phase shift is introduced in the periodic term to meet a hidden solvability condition lurking within the problem.  In the second, the phase shift is fixed as a (more or less) arbitrary value, and now the periodic amplitude is selected to satisfy the solvability condition.  These constructions adapt different techniques due to Beale and Lombardi for related systems and is intended as the first step in a broader program uniting the flexible framework of Beale’s methods with the precision of Lombardi’s for applications to various problems in lattice dynamical systems.  As a more immediate application, we use the results for the model problem to solve a system of coupled KdV-KdV equations that models the propagation of certain surface water waves.

A deter-mean-istic description of Stochastic Oscillators

Series
CDSNS Colloquium
Time
Friday, May 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Alberto Pérez-CerveraUniversidad Complutense de Madrid, Spain

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

Abstract: The Parameterisation Method is a powerful body of theory to compute the invariant manifolds of a dynamical system by looking for a parameterization of them in such a way that the dynamics on this manifold expressed in the coordinates of such parameterization writes as simply as possible. This methodology was foreseen by Guillamon and Huguet [SIADS, 2009 & J. Math. Neurosci, 2013] as a possible way of extending the domain of accuracy of the phase-reduction of periodic orbits. This fruitful approach, known as phase-amplitude reduction, has been fully developed during the last decade and provides an essentially complete understanding of deterministic oscillatory dynamics.
In this talk, we pursue the "simpler as possible" philosophy underlying the Parameterisation Method to develop an analogous phase-amplitude approach to stochastic oscillators. Main idea of our approach is to find a change of variables such that the system, when transformed to these variables, expresses in the mean as the deterministic phase-amplitude description. Then, we take advantage of the simplicity of this approach, to develop interesting objects with the aim of further clarifying the stochastic oscillation.

Free energy and uniqueness in 1D spin systems with random Hamiltonians

Series
CDSNS Colloquium
Time
Friday, April 28, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Cesar Octavio Maldonado AhumadaIPICYT


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

Abstract: In this talk, I will discuss problems and results in the rigorous statistical mechanics of particle systems in a one-dimensional lattice.
I will briefly describe the classical examples, such as the Ising model and its various generalizations concerning the
existence of the free energy, thermodynamic limit and the phase transition phenomenon.
Towards the end of the talk, I will talk about a recent work in collaboration with Jorge Littin, on a generalization of the
Khanin and Sinai model with random interactions for which one can prove that there exists a critical behavior in the free
energy for some parameters of the model and on the other side one can also have uniqueness of the equilibrium state.


 

Bifurcations in patterns of human sleep under variation in homeostatic dynamics

Series
CDSNS Colloquium
Time
Friday, April 21, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and online
Speaker
Christina AthanasouliGeorgia Tech

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

Abstract: The timing of human sleep is strongly modulated by the 24 hour circadian rhythm, our internal biological clock, and the homeostatic sleep drive, one’s need for sleep which depends on prior awakening. The parameters dictating the evolution of the homeostatic sleep drive may vary with development and have been identified as important parameters for generating the transition from multiple sleeps to a single sleep episode per day. We employ piecewise-smooth ODE-based mathematical models to analyze developmentally-mediated transitions of sleep-wake patterns, including napping and non-napping behaviors. Our framework includes the construction of a circle map that captures the timing of sleep onsets on successive days. Analysis of the structure and bifurcations in the map reveals changes in the average number of sleep episodes per day in a period-adding-like structure. In two-state models of sleep-wake regulation, namely models that generate sleep and wake states, we observe saddle-node and border collision bifurcations in the maps. However, in our three-state model of sleep-wake regulation, which captures wake, rapid eye movement (REM) sleep, and non-REM sleep, these sequences are disrupted by period-doubling bifurcations and can exhibit bistability.

Toward algorithms for linear response and sampling

Series
CDSNS Colloquium
Time
Friday, April 14, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Nisha ChandramoorthyGeorgia Tech

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

Abstract: Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems. This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle's formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold.

In the second part, we discuss a new KAM method-inspired construction of transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.

Self-similar blow up profiles for fluids via physics-informed neural networks

Series
CDSNS Colloquium
Time
Friday, April 7, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and online
Speaker
Javier Gomez SerranoBrown University

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

Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.

Joint work with Yongji Wang, Ching-Yao Lai and Tristan Buckmaster.

Hill Four-Body Problem with oblate bodies

Series
CDSNS Colloquium
Time
Friday, March 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Wai Ting LamFAU

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

G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor. 

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.

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