Investigating the phase space dynamics of conservative dynamical systems by the Lagrangian descriptors method
In this work, we numerically investigate the dynamics of conservative dynamical systems using the method of Lagrangian descriptors (LDs), which has been extensively used to visualise characteristic features (like fixed points, periodic orbits and their associated manifolds) in the phase space of nonlinear dynamical systems. The computation of LDs is based on the accumulation of a positive scalar value along any orbit of the dynamical system, making them a rather easily evaluated quantity. Firstly, we use the method of LDs to study the escape of stars in an analytic model of a rotating galaxy. We find that the LDs are able to visually describe the lobe structure of manifolds which govern the escape of stars, thereby providing a computationally cheap and simple way to depict and analyse the model's phase space structures. Secondly, we develop and validate chaos detection techniques which use computations of the LDs of nearby orbits, to characterise the chaoticity of generic conservative systems. More specifically, we introduce the difference and ratio of the LDs of neighbouring orbits as chaos detection diagnostics, and include in our study a quantity related to the second spatial derivative of LDs, which was recently developed by other researchers. Applying these techniques to three basic, prototypical models, namely the two degrees of freedom H´enon-Heiles system, the two-dimensional (2D) standard map, and the 4D standard map, we find that these indices identify chaotic orbits with an accuracy of ≳ 90% when compared to the Smaller Alignment Index (SALI) method, which is a well-established chaos detection technique. Furthermore, we determine the effect on the indices' performance of (i) the orbits' integration time, (ii) the grid spacing between the considered neighbouring orbits, (iii) the number and arrangement of the nearby orbits used to evaluate the indicators and (iv) the overall extent of chaos in the system. A basic outcome of our work is that these indicators can be used to efficiently characterise chaotic behaviour of both low and high-dimensional dynamical systems at short integration times, without solving the so-called variational equations for continuous time systems, or evaluating the tangent map for discrete time models, needed by other, traditional chaos detection techniques.
Networked Digital Library of Theses and Dissertations
Department of Mathematics and Applied Mathematics