Georgia Institute of Technology College of Sciences

We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.

Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate (Arnold diffusion). 

This can be understood by noting that the small perturbations generate some invariant objects in phase space that act as routes which allow accumulation of effects. 

We will present some rigorous results about geometric objects lead to Arnold diffusion as well as some computational tools that allow to find them in concrete applications.

Thanks to the work of many people, an area which used to be very speculative, is becoming an applicable tool.

Starting from mathematical approaches for image processing, we will discuss different models, analytic aspects of them, and numerical challenges.  If time permits we will consider numerical applications to data understanding. A few other applications may be presented.

One of the simplest and, at the same time, most prominent models for the discrete quasi-periodic Schrodinger operator is the almost Mathieu operator (also called the Harper's model). This simple-looking operator is known to present exotic spectral properties. Three (out of fifteen) of Barry Simon's problems on Schrodinger operators in the 21st century concerns the almost Mathieu operator. In 2014, Artur Avila won a Fields Medal for work including the solutions to these three problems. In this talk, I will concentrate on the one concerning the Lebesgue measure of the spectrum. I will also talk about the difficulties in generalizing this result to the extended Harper's model. Students with background in numerics are especially welcome to attend!

Basin of attraction for a stable equilibrium point is an effective concept for stability in deterministic systems. However, it does not contain information on the external perturbations that may affect it. The concept of stochastic basin of attraction (SBA) is introduced by incorporating a suitable probabilistic notion of basin. The criteria for the size of the SBA is based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small intensity but also with whose amplitude is proportional or in general is a function of an order parameter. The efficiency of the concept is presented through two applications.