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Complex Dynamics of Magnetic Billiards in a 2-Torus

Silvans, Albert (2023) Complex Dynamics of Magnetic Billiards in a 2-Torus. Master's Thesis / Essay, Mathematics.


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The study of mathematical billiards is a prominent topic within dynamical systems. One of the most important examples of a billiard is the Sinai billiard: a single particle on a 2-torus with a circular obstacle, if the particle collides with the boundary of the obstacle, it does so elastically. Once the particle is set in motion, its long term behavior is studied: is the motion ergodic, do there exist periodic trajectories? Other such billiards can be devised, and for each we can expect its own challenge. We consider a variation of the Sinai billiard where the solid boundary is removed and in the interior of the disc there is a magnetic field. Now, the particle passes through and its trajectory is deflected. This introduces a parameter, the magnetic field strength, which we can vary, and study. We approach the magnetic Sinai billiard from two perspectives. First, we consider it in the context of KAM theory, which deals with Hamiltonian systems and small perturbations. Next, we study the system using symbolic dynamics, more specifically, we use the Lempel-Ziv complexity to distinguish initial conditions that lead to ordered dynamics from those that lead to disorder. In such a way, we characterize qualitatively different dynamics of the system.

Item Type: Thesis (Master's Thesis / Essay)
Supervisor name: Seri, M. and Jardon Kojakhmetov, H.
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 22 Nov 2023 08:00
Last Modified: 22 Nov 2023 08:00

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