טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentPinsky Tali
SubjectCharacterization of Periodic Orbits by Topological
Considerations for a Kicked Particle
DepartmentDepartment of Mathematics
Supervisors Professor Emeritus Bronislaw Wajnryb
Professor Emeritus Shmuel Fishman (Deceased)


Abstract

We give an order relation on a certain family of periodic orbits which exist for a certain type of homeomorphisms of the torus called shear type homeomorphisms. Each orbit in this family is characterized by its rotation number, which is some rational number in the unit interval. The order relation is then induced by a natural order relation on rational numbers and pairs of rational numbers which are Farey neighbors. The work is thus an analog of Sharkovskii’s theorem for a specific two-dimensional case.
The topological tools used in order to achieve this are the Bestvina-Handel algorithm, Markov partitions and theorems regarding isotopy stability of periodic orbits. The results are then used for analyzing a concrete physical system, which is a one-dimensional system (hence with a two-dimensional phase space). The system is a quantum chaotic system called the kicked accelerated particle, consisting of non-interacting particles subject to gravitation and kicked by a periodic electromagnetic field. This system can be modeled as a map on the two-torus which is of shear type, and the periodic orbits found for the map have physical significance as they give rise to accelerator modes. The order relation found then gives a global qualitative understanding of the structure of existence of these periodic orbits, and so existence of accelerator modes, depending on the parameters of the physical system.