Georgia Institute of Technology College of Sciences

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.

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.

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.

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.