Seminars and Colloquia by Series

Conditioned Random Dynamics and Quasi-ergodic measures

Series
CDSNS Colloquium
Time
Friday, November 10, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Matheus de CastroImperial College

In this talk, we study the long-term behaviour of Random Dynamical Systems (RDSs) conditioned upon staying in a region of the space. We use the absorbing Markov chain theory to address this problem and define relevant dynamical systems objects for the analysis of such systems. This approach aims to develop a satisfactory notion of ergodic theory for random systems with escape.

Is Nambu mechanics a generalization of Hamiltonian mechanics?

Series
CDSNS Colloquium
Time
Friday, November 3, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Cristel ChandreGeorgia Tech

In 1973, Nambu published an article entitled "Generalized Hamiltonian dynamics". For that purpose, he constructed multilinear brackets - equivalent to Poisson brackets - with some interesting properties reminiscent of the Jacobi identity.
These brackets found some applications in fluid mechanics, plasma physics and mathematical physics with superintegrable systems.

In this seminar, I will recall some basic elements on Nambu mechanics in finite dimension with an n-linear Nambu bracket in dimension larger than n. I will discuss all possible Nambu brackets and compare them with all possible Poisson brackets. I will conclude that Nambu mechanics can hardly be considered a generalization of Hamiltonian mechanics.

Arnold Tongues in Standard Maps with Drift

Series
CDSNS Colloquium
Time
Friday, October 27, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Jing ZhouGreat Bay University

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.

Computation of high-order normal forms in diffeomorphisms

Series
CDSNS Colloquium
Time
Friday, October 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Joan GimenoGeorgia Tech

This talk will delve into a method specifically designed for
constructing high-order normal forms in Poincaré maps with high-order
precision and without any major assumption or structure of the
dynamical system itself. We will use the result to generate explicit
twist maps, calculating invariant tori, and determining the flying
time expansions around an elliptic fixed point of a Poincaré map. In
particular, this approach is able to check some non-degenerate
conditions in perturbation theory.

A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem

Series
CDSNS Colloquium
Time
Friday, October 6, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249 (in-person)
Speaker
Jaime ParadelaUniversity of Maryland

A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0, m_1$. In the region of the phase space where the massless body is far from the primaries, the problem can be studied as a (fast) periodic perturbation of the 2 Body Problem (2BP), which is integrable.

We prove that the restricted 3BP exhibits topological instability: for any values of the masses $m_0, m_1$ (except $m_0 = m_1$), we build orbits along which the angular momentum of the massless body (conserved along the flow of the 2BP) experiences an arbitrarily large variation. In order to prove this result we show that a degenerate Arnold diffusion mechanism takes place in the restricted 3BP. Our work extends previous results by Delshams, Kaloshin, De la Rosa and Seara for the a priori unstable case $m_1< 0$, where the model displays features of the so-called a priori stable setting. This is joint work with Marcel Guardia and Tere Seara.

Arnold diffusion in Hamiltonian systems with small dissipation

Series
CDSNS Colloquium
Time
Monday, October 2, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 005
Speaker
Marian GideaYeshiva University

We consider a mechanical system consisting of a rotator and a pendulum, subject to a small, conformally symplectic perturbation. The resulting system has energy dissipation. We provide explicit conditions on the dissipation parameter, so that the resulting system exhibits Arnold diffusion. More precisely, we show that there are diffusing orbits along which the energy of the rotator grows by an amount independent of the smallness parameter. The fact that Arnold diffusion may play a role  in  systems with small dissipation was conjectured by Chirikov. Our system can be viewed as a simplified  model for an energy harvesting device, in which context the energy growth translates into generation of electricity.
Joint work with S.W. Akingbade and T-M. Seara.

Topological dynamics of knotted and tangled matter

Series
CDSNS Colloquium
Time
Friday, September 29, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Vishal PatilStanford
Knots and tangles play a fundamental role in the dynamics of biological and physical systems, from DNA and root networks to surgical sutures and shoelaces. Despite having been studied for centuries, the subtle interplay between topology and mechanics in tangled elastic filaments remains poorly understood. Here we investigate the dynamical rules governing the behavior of knotted and tangled matter. We first study the human-designed knots used to tie ropes together. By developing an analogy with long-range ferromagnetic spin systems, we identify simple topological counting rules to predict the relative mechanical stability of commonly used climbing and sailing knots. Secondly, we examine the complex tangling dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using ultrasound imaging datasets, we construct a minimal model that explains how the kinematics of individual active filaments determines their emergent collective topological dynamics. By identifying generic dynamical principles of topological transformations, our results can provide guidance for designing classes of self-adaptive topological metamaterials.

 

 

 

 

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.


 

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