Browsing by Author "Skokos, Charalampos"

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    Chaotic behavior and energy polarisation in flatband lattice models
    (2023) Cheong, Su Ho; Skokos, Charalampos
    Flatbands (FBs) in lattice systems correspond to dispersionless energy bands cre- ated through what is called destructive interference, a phenomenon caused by the existence of lattice symmetries, resulting in compactly localised wave functions to just a few lattice sites. The existence of FBs in materials entails high sensitivity to the initial conditions of the system, especially with regard to disorder and nonlinear- ity. In this study, we numerically investigate the wave packet dynamics and chaotic behaviour of a simple tight-binding system exhibiting FBs, the so-called stub lattice model. Initially we show the existence of a FB and of two dispersive frequency bands for this model and identify three different dynamical regimes, namely the weak chaos, strong chaos, and self trapping regimes. Using symplectic integration techniques, we evolve in time, t, initially localised wave packets for these regimes and quantify their spreading through the computation of the wave packets second moment, m2, while the extent of localisation is characterised using the wave packets participation number. We show that, for both the weak and strong chaos regimes, the spreading of wave packets, which is characterised by a power law increases of m2 (respectively ∝ t0.33 and ∝ t0.5), is a chaotic process whose maximum Lyapunov exponent respectively decreases ∝ t−0.25 and ∝ t−0.3. By decreasing the system's disorder strength we alter the width of the bandgap, something which does not ap- pear to affect the spreading dynamics in the weak chaos regime, while our results do not lead to clear conclusions in the case of strong chaos. Furthermore, we find that particular disorder configurations which preserve the FB do not alter the wave packet spreading dynamics for the weak chaos regime. Finally, we observe that the wave packets norm distributions at the subsites of the stub lattice unit cells reach equilibrium if the evolution time is sufficiently long
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    Investigating chaos by the generalized alignment index town (GALI) method
    (2020) Moges, Henok Tenaw; Skokos, Charalampos
    One of the fundamental tasks in the study of dynamical systems is the discrimination between regular and chaotic behavior. Over the years several methods of chaos detection have been developed. Some of them, such as the construction of the system's Poincar´e Surface of Section, are appropriate for low-dimensional systems. However, an enormous number of real-world problems are described by high-dimensional systems. Thus, modern numerical methods like the Smaller (SALI) and the Generalized (GALI) Alignment Index, which can also be used for lower-dimensional systems, are appropriate for investigating regular and chaotic motion in high-dimensional systems. In this work, we numerically investigate the behavior of the GALIs in the neighborhood of simple stable periodic orbits of the well-known Fermi-Pasta-Ulam-Tsingou lattice model. In particular, we study how the values of the GALIs depend on the width of the stability island and the system's energy. We find that the asymptotic GALI values increase when the studied regular orbits move closer to the edge of the stability island for fixed energy, while these indices decrease as the system's energy increases. We also investigate the dependence of the GALIs on the initial distribution of the coordinates of the deviation vectors used for their computation and the corresponding angles between these vectors. In this case, we show that the final constant values of the GALIs are independent of the choice of the initial deviation vectors needed for their computation.
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    Investigating the phase space dynamics of conservative dynamical systems by the Lagrangian descriptors method
    (2023) Zimper, Sebastian; Skokos, Charalampos
    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.
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    Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
    (2021) Many, Manda Bertin; Skokos, Charalampos
    In this thesis we study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. Recent works focussing on two widely–applicable systems, namely the disordered Klein-Gordon (DKG) lattice of anharmonic oscillators and the disordered discrete nonlinear Schr¨odinger (DDNLS) equation, mainly in one spatial dimension suggest that nonlinearity eventually destroys AL. This leads to an infinite diffusive spreading of initially localized wave packets whose extent (measured for instance through the wave packet's second moment m2) grows in time t as t αm with 0 < αm < 1. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. Two different spreading regimes, the so-called weak and strong chaos regimes, have been theoretically predicted and numerically identified. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the system's tangent dynamics. In particular, we consider the DDNLS model of one (1D) and two (2D) spatial dimensions and develop robust, efficient and fast numerical integration schemes for the long-time evolution of the phase space and tangent dynamics of these systems. Implementing these integrators, we perform extensive numerical simulations for various sets of parameter values. We present, to the best of our knowledge for the first time, detailed computations of the time evolution of the system's maximum Lyapunov exponent (MLE–Λ) i.e. the most commonly used chaos indicator, and the related deviation vector distribution (DVD). We find that although the systems' MLE decreases in time following a power law t αΛ with αΛ < 0 for both the weak and strong chaos cases, no crossover to the behavior Λ ∝ t −1 (which is indicative of regular motion) is observed. By investigating a large number of weak and strong chaos cases, we determine the different αΛ values for the 1D and 2D systems. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites. Furthermore, we show the existence of a dimension-free relation between the wave packet spreading and its degree of chaoticity between the 1D and 2D DDNLS systems. The generality of our findings is confirmed, as similar behaviors to the ones observed for the DDNLS systems are also present in the case of DKG models.