Hamiltonian chaos: from galactic dynamics to plasma physics

Tenaw , Henok

Hamiltonian chaos: from galactic dynamics to plasma physics

Thesis / Dissertation

2025

Publisher

University of Cape Town

Department

Department of Mathematics and Applied Mathematics

Faculty

Faculty of Science

Abstract

The primary focus of this thesis is the numerical investigation of the influence of chaos in Hamiltonian models describing the behavior of charged particle orbits in plasma, the motion of stars in barred galaxies, and the diffusion of trajectories in multidimensional maps. First, we systematically explore the interplay between magnetic and kinetic chaos in toroidal fusion plasmas, where non-axisymmetric perturbations disrupt smooth magnetic flux surfaces, generating complex particle trajectories. Using the Generalized Alignment Index (GALI) method of chaos detection, we efficiently quantify chaos, compare the behavior of magnetic field lines and particle orbits, visualize the radial distribution of chaotic regions, and offer the GALI method as a valuable tool for studying the dynamics of plasma physics models. Next, we study the evolution of phase space structures in a three-dimensional (3D) barred galactic potential, following successive 2D and 3D pitchfork and period-doubling bifurcations of periodic orbits. By employing the so-called 'color and rotation' technique to visualize the four-dimensional Poincare surface ´ of sections of the system, we reveal distinct structural patterns. We further investigate the long-term diffusion transport and chaos properties of single and coupled standard maps, focusing on parameters that induce anomalous diffusion through the presence of accelerator modes exhibiting ballistic transport. Using different ensembles of initial conditions in chaotic regions influenced by these modes, we examine asymptotic diffusion rates and their corresponding time scales, identifying conditions that suppress anomalous transport and lead to long-term convergence to normal diffusion across various coupled map arrangements. Lastly, we perform the first comprehensive investigation into the behavior of the GALI indices for various attractors in continuous and discrete-time dissipative systems, extending the application of the method to non-Hamiltonian systems. A key aspect of our work involves analyzing and comparing the performance of the GALI method with the computation of Lyapunov Exponents for non-Hamiltonian dissipative systems exhibiting hyperchaotic motion.

Keywords

applied mathematics

Reference:

Collections

PhD / Doctoral

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