Diffusion and transport in Hamiltonian systems

Lead Research Organisation: Imperial College London
Department Name: Dept of Mathematics

Abstract

The Arnold diffusion is a major mechanism of instability in dynamics of multidimensional systems. Obtaining a rigorous mathematical description of this process is one of the main problems of the theory of Hamiltonian dynamics, which is still unresolved. The goal of the project is to create new geometrical methods for the study and, possibly, a solution to this problem.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509486/1 01/10/2016 31/03/2022
1831857 Studentship EP/N509486/1 01/10/2016 31/03/2020 Andrew Clarke
 
Description It was shown that a generic strictly convex billiard domain in 3 dimensions or more on which the geodesic flow has a hyperbolic periodic orbit with a transverse homoclinic contains billiard trajectories where the angle of reflection tends to zero. Such trajectories were first considered in the early 1990's, but there exist was unknown before now. Next, it was shown that generic strictly convex surfaces in 3-dimensional Euclidean space have a hyperbolic closed geodesic with a transverse homoclinic.
Exploitation Route These results lay a foundation for future proofs of generic existence of transverse homoclinic geodesics on hypersurfaces. Moreover, the results extend what is known regarding real-analytic perturbation theory.
Sectors Aerospace, Defence and Marine,Education,Culture, Heritage, Museums and Collections

URL https://arxiv.org/abs/1906.07778