Georgia Institute of Technology College of Sciences

We investigate dynamical systems obtained by coupling  an Anosov diffeomorphism and a N-pole-to-S-pole map of the circle. Both maps are uniformly hyperbolic; however, they have contrasting character, as the first one is chaotic while the second one has “orderly" dynamics. The first thing we show is that even weak coupling can produce interesting phenomena: when the attractor of the uncoupled system is not normally hyperbolic, most small interactions transform it from a smooth surface to a fractal-like set.  We then consider stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. These couplings produce genuine obstructions to uniform hyperbolicity; however, the monotonicity conditions make the system amenable to study by leveraging  techniques from the geometric and ergodic theories of hyperbolic systems.  In particular, we can show existence of invariant cones and SRB measures. 

This is joint work with Lai-Sang Young.

Delay differential equations (DDEs) are important in physical applications where there is a time lag in communication between subsystems.  From a mathematical point of view DDEs are an interesting source of problems as they provide natural examples of infinite dimensional dynamical systems.  I'll discuss some spectral numerical methods for computing invariant manifolds for DDEs and present some applications.  

Riemann's non-differentiable function, although introduced as a pathological example in analysis, makes an appearance in a certain limiting regime of the theory of binormal flow for vortex lines. From this physical point of view, it also bears some qualitative similarities to turbulent fluid velocity fields in the infinite Reynolds number limit. In this talk, we'll see how this function arises in the study of the vortex filaments, and how we can adapt the notion of intermittency from the study of turbulent flows to this setting. Then, we'll study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways. One in the physical space, and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0, which highlights the (weak) intermittent nature of Riemann's function.

This is a joint work with Alexandre Boritchev (Université de Lyon) and Daniel Eceizabarrena (BCAM, Bilbao).
 

We will present the "a-posteriori" approach to KAM theory.

We formulate an invariance equation and show that an approximate-enough solution which verifies some non-degeneracy conditions leads to a solution.  Note that this does not have any reference to integrable systems and that the non-degeneracy conditions are not global properties of the system, but only properties of the solution. The "automatic reducibility" allows to take advantage of the geometry to develop very efficient Newton methods and show that they converge.

This leads to very efficient numerical  algorithms (which moreover can be proved to lead to correct solutions), to validate formal expansions. From a more theoretical point of view, it can be applied to other geometric contexts (conformally symplectic, presymplectic) and other geometric objects such as whiskered tori. One can deal well with degenerate systems, singular perturbation theory and obtain simple proofs of monogenicity and Whitney regularity.

This is joint work with many people.