Browsing by Author "Skokos, Haris"

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    Application of the Lagrangian descriptors method to Hamiltonian systems with emphasis to models of barred galaxies
    (2024) Theron, Dylan Grant; Skokos, Haris
    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.
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    Chaotic behaviour of charged particles in electromagnetic fields
    (2018) Ani, Chinenye Jane; Skokos, Haris
    In order to understand the motion of charged particles we numerically investigate the chaoticity of magnetic field lines of tokamak fields, as charged particles move along field lines. In particular, the symmetric tokamap was studied to determine the physical quantities that influence the system’s chaotic behaviour. We implement several chaos detection techniques: the construction of Poincaré maps, the computation of the maximum Lyapunouv characteristic exponent (mLCE), as well as the Smaller Alignment Index (SALI). The analyses performed showed that the mLCE and SALI methods accurately quantified magnetic field lines’ chaotic behaviour and that the relative perturbation strength influences the system’s chaoticity. In addition, we illustrate the diffusive properties of magnetic field lines, using statistical measures like the mean square displacement (MSD) and calculating diffusion coefficients. Lastly, we present the construction of explicit near-symplectic mappings of the symmetric tokamap with Lie-generating functions.
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    Chaotic behaviour of disordered nonlinear lattices
    (2021) Senyange, Bob; Skokos, Haris
    In this work we systematically investigate the chaotic energy spreading in prototypical models of disordered nonlinear lattices, the so-called disordered Klein-Gordon (DKG) system, in one (1D) and two (2D) spatial dimensions. The normal modes' exponential localization in 1D and 2D heterogeneous linear media explains the phenomenon of Anderson Localization. Using a modified version of the 1D DKG model, we study the changes in the properties of the system's normal modes as we move from an ordered version to the disordered one. We show that for the ordered case, the probability density distribution of the normal modes' frequencies has a ‘U'-shaped profile that gradually turns into a plateau for a more disordered system, and determine the dependence of two estimators of the modes' spatial extent (the localization volume and the participation number) on the width of the interval from which the strengths of the on-site potentials are randomly selected. Furthermore, we investigate the numerical performance of several integrators (mainly based on the two part splitting approach) for the 1D and 2D DKG systems, by performing extensive numerical simulations of wave packet evolutions in the various dynamical regimes exhibited by these models. In particular, we compare the computational efficiency of the integrators considered by checking their ability to correctly reproduce the time evolution of the systems' finite time maximum Lyapunov exponent estimator Λ and of various features of the propagating wave packets, and determine the best-performing ones. Finally we perform a numerical investigation of the characteristics of chaos evolution for a spreading wave packet in the 1D and 2D nonlinear DKG lattices. We confirm the slowing down of the chaotic dynamics for the so-called weak, strong and selftrapping chaos dynamical regimes encountered in these systems, without showing any signs of a crossover to regular behaviour. We further substantiate the dynamical dissimilarities between the weak and strong chaos regimes by establishing different, but rather general, values for the time decay exponents of Λ. In addition, the spatio-temporal evolution of the deviation vector associated with Λ reveals the meandering of chaotic seeds inside the wave packets, supporting the assumptions for chaotic spreading theories of energy.
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    Chaotic Dynamics of Polyatomic Systems with an Emphasis on DNA Models
    (2021) Hillebrand, Malcolm; Skokos, Haris
    In this work we investigate the chaotic behaviour of multiparticle systems, and in particular DNA and graphene models, by applying various numerical methods of nonlinear dynamics. Through the use of symplectic integration techniques—efficient routines for the numerical integration of Hamiltonian systems—we present an extensive analysis of the chaotic behaviour of the Peyrard-Bishop-Dauxois (PBD) model of DNA. The chaoticity of the system is quantified by computing the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair disorder on the dynamics is investigated. In addition to the inherent heterogeneity due to the proportion of adenine-thymine (AT) and guanine-cytosine (GC) base pairs, the distribution of these base pairs in the sequence is analysed through the introduction of the alternation index. An exact probability distribution for arrangements of base pairs and their alternation index is derived through the use of Pólya counting theory. We find that the value of the mLE depends on both the base pair composition of the DNA strand and the arrangement of base pairs, with a changing behaviour depending on the temperature. Regions of strong chaoticity are probed using the deviation vector distribution, and links between strongly nonlinear behaviour and the formation of bubbles (thermally induced openings) in the DNA strand are studied. Investigations are performed for a wide variety of randomly generated sequences as well as biological promoters. Furthermore, the properties of these thermally induced bubbles are studied through large-scale molecular dynamics simulations. The distributions of bubble lifetimes and lengths in DNA are obtained and discussed in detail, fitted with simple analytical expressions, and a physically justified threshold distance for considering a base pair to be open is proposed and successfully implemented. In addition to DNA, we present an analysis of the dynamical stability of a planar model of graphene, studying the behaviour of the mLE in bulk graphene sheets as well as in finite width graphene nanoribbons (GNRs). The wellattested stability of the material manifests in a very small mLE, with chaos being a slow process in graphene. For both possible kinds of GNR, armchair and zigzag edges, the mLE decreases with increasing width, asymptotically reaching the bulk behaviour. This dependence of the mLE on both energy density and ribbon width is fitted accurately with empirical expressions.