Chaotic behaviour of disordered nonlinear lattices

Senyange, Bob

Chaotic behaviour of disordered nonlinear lattices

Doctoral Thesis

2021

Abstract

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.

Keywords

Mathematics and Applied Mathematics

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PhD / Doctoral

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