Unstable Dynamics in Hamiltonian Systems

The key challenge of the modern theory of Hamiltonian Dynamical Systems is to provide adequate mathematical tools for describing chaotic dynamics in a system which combines both regular and chaotic components as an overwhelming majority of models relevant for applications of the theory fell into this category. In this area two major problems have resisted the efforts of scientists for decades. The first one is called "Arnold Diffusion" and is related to instability of action variables on long time scales. The second one is known as "Positive Metric Entropy Conjecture" and states that chaotic motions are physically relevant, i.e., they occupy a subset of positive Lebesgue measure.

In the period of the fellowship I will concentrate on the study of Hamiltonian systems with multiple time scales, as this property is often present in the equations either explicitly or implicitly. The aim of this study is to develop mathematical tools for studying instabilities of dynamics. Preliminary results show that we are able to prove existence of normally hyperbolic invariant objects with different dynamical behaviour of slow components. It is probable that on longer time scales the restriction of the dynamical system on this family can be approximated by a stochastic ordinary differential equation in the slow variables. If confirmed, it will establish an important connection between two different fields of Mathematics: the theory of deterministic Hamiltonian systems and stochastic differential equations, which are considered mostly unrelated at the present.

An extension of these results should provide a new insight on the theory of the Fermi acceleration.

A comparison with variational approach to Arnold Diffusion announced by Mather (Princeton) suggests that our mechanism could be used to solve the long-standing problem of genericity of Arnold Diffusion in near-integrable Hamiltonian systems. The progress in this direction should require radical improvements of methods for detection of transversal homoclinic trajectories associated with various invariant objects, i.e., in the area where I have a substantial technical expertise.

As a summary, the following list of technical topics will be addressed initially: exponentially small splitting of invariant manifolds in higher dimension, stochastic description of slow dynamics in slow-fast systems with chaotic fast component, Fermi acceleration, Arnold Diffusion, positive metric entropy conjecture.

I expect that as the fellowship advances new research directions will arise partially motivated by the development of the theory and partially by questions coming from its applications.

The research will be curried out at the Mathematics Institute, University of Warwick, and will involve collaboration with several groups in the UK and oversees.

The primary beneficiaries will be other researchers in related areas. The results of the proposed research will potentially lead to better understanding of celestial mechanics, some processes in plasma physics, and other areas of mathematical and theoretical physics which use Hamiltonian equations for modelling, and also affect understanding of mathematical background of Statistical Physics. Although the main goal of the proposed research is related to understanding the fundamental principles and direct technical applications do not form a part of the proposals, it is feasible that, in longer run, the theory may have practical applications, e.g., it can be useful for development of devices capable of fine tuning a laser frequency or for conducting an analysis of dynamics of small bodies in the Solar System. The project will also contribute to development of the human resources by providing training to a PhD student and a young postdoctoral scientist in the advanced theory of Hamiltonian systems.