Hamiltonian Chaos: From Galactic Dynamics to Plasma Physics

TL;DR

This work systematically investigates chaos in Hamiltonian and dissipative dynamical systems using the Generalized Alignment Index (GALI) and Lyapunov exponents. It demonstrates GALI’s efficiency in distinguishing regular from chaotic motion across plasma, galactic, and map-based models, and extends its application to non-Hamiltonian dissipative systems. Key contributions include a first application of GALI to magnetic and kinetic chaos in toroidal plasmas, a detailed analysis of phase-space bifurcations in a 3D bar galaxy potential, and comprehensive diffusion studies in single and coupled standard maps, highlighting accelerator-mode–driven anomalous diffusion and coupling-induced normal diffusion. The findings illuminate how chaos indicators relate to transport properties, phase-space structure, and stability transitions, with practical implications for confinement in fusion devices and understanding galactic dynamics. The work also benchmarks GALI against conventional Lyapunov methods, clarifying its strengths in speed and sensitivity for a broad class of dynamical systems and identifying limitations in certain regular- versus chaotic-attractor distinctions in dissipative regimes.

Abstract

The primary focus of this thesis is the numerical investigation of chaos in Hamiltonian models describing charged particle orbits in plasma, star motions in barred galaxies, and orbits' diffusion in multidimensional maps. 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, we efficiently quantify chaos, compare the behavior of magnetic field lines and particle orbits, visualize the radial distribution of chaotic regions, and offer GALI as a valuable tool for studying plasma physics dynamics. We also study the evolution of phase space structures in a 3D barred galactic potential, following successive 2D and 3D pitchfork and period-doubling bifurcations of periodic orbits. By employing the `color and rotation' technique to visualize the system's 4D Poincaré surface of sections, 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 inducing anomalous diffusion through accelerator modes exhibiting ballistic transport. Using different ensembles of initial conditions in chaotic regions influenced by these modes, we examine asymptotic diffusion rates and time scales, identifying conditions suppressing anomalous transport and leading to long-term convergence to normal diffusion across coupled maps. Lastly, we perform the first comprehensive investigation into the GALI indices for various attractors in continuous and discrete-time dissipative systems, extending the method's application to non-Hamiltonian systems. A key aspect of our work involves analyzing and comparing GALIs' with Lyapunov Exponents for systems exhibiting hyperchaotic motion.

Figures (73)

• Figure 1: Basic structure of a Tokamak [image source: www.euro-fusion.org].
• Figure 2: The general toroidal coordinate system used to describe the MF surfaces in a toroidal fusion device WhiteBook.
• Figure 3: Toroidal surface (constant $\zeta$) and poloidal surface (constant $\theta$) defining the magnetic flux $\psi$ and $\psi_p$, respectively. Both $\psi$ and $\psi_p$ are zero at the magnetic axis (gray region) WhiteBook.
• Figure 4: The PSS $(\theta = 0; p_\theta > 0)$ of the GC Hamiltonian \ref{['eq:GCM-Ham-q=1']} for normalized energy $E = 8.131 \times 10^{-6}$ and magnetic moment $\mu = 8.1423\times10^{-6}$, subject to perturbations with mode numbers $(m_1, n_1) = (1, 5)$, $(m_2, n_2) = (1, 3)$, and perturbation amplitude $\epsilon = 0.135\times10^{-8}$. Regular orbits correspond to blue, orange, and purple points, while chaotic orbits are shown by green, red, and black points. The ICs for the chaotic orbits are $p_{\zeta_0} = -0.85 \times 10^{-3}$ (red square point), $p_{\zeta_0} = -0.81 \times 10^{-3}$ (green triangle point), and $p_{\zeta_0} = -0.36 \times 10^{-3}$ (black diamond point) with $\zeta = 0$.
• Figure 5: The time evolution of (a) the ftmLE, $\sigma_1$\ref{['eq:ftmLE']}, and (b) the GALI$_2$\ref{['eq:GALI']} for the six GC orbits shown in Fig. \ref{['fig3:Fig1']} using the same color scheme. The dashed line in (a) represents a function proportional to $t^{-1}$.