Application of the Lagrangian descriptors method to Hamiltonian systems with emphasis to models of barred galaxies - Theron, Dylan Grant (Department of Mathematics and Applied Mathematics, 2025)
The Lagrangian descriptors (LDs) method is a numerical technique that assigns to an orbit's initial condition a positive scalar value. Its implementation permits the conversion of a dynamical system's phase space into a scalar field which can be used to distinguish regions of different dynamical behaviours and ultimately reveal structures in the system's phase space. In this work, we apply the LDs method to different dynamical systems. We first study a Hamiltonian system of galactic type to highlight normally hyperbolic invariant manifolds (NHIMs), examining the impact of different pattern speeds and energy levels on the NHIMs' structure and determine how these features influence orbital morphologies seen in the model's configuration space. Thereafter, we apply the LDs method to a dynamical system whose evolution is governed by fractional ordinary differential equations (FDEs) and showcase the utility of this method in qualitatively revealing phase space structures for systems described by FDEs. In our study, we implement two numerical techniques to integrate such systems, namely the Grunwald-Letnikov (GL) method to solve Caputo type derivatives and the GL approximation for Riemann-Liouville derivatives. We emphasise the differences between these two methods and examine the resulting phase space structures. Additionally, we investigate the effect of the final integration time and the order of the involved fractional derivatives on the features seen in the system's phase portraits, which are revealed through the computation of the LDs for large ensembles of orbits.