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Register a new userNonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
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Authors
Bertin Many Manda
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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. 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. 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 systems tangent dynamics. In particular, we consider the disordered discrete nonlinear Schrodinger (DDNLS) equation of one (1D) and two (2D) spatial dimensions. We present detailed computations of the time evolution of the systems maximum Lyapunov exponent (MLE--$Lambda$), and the related deviation vector distribution (DVD). We find that although the systems MLE decreases in time following a power law $t^{alpha_Lambda}$ with $alpha_Lambda <0$ for both the weak and strong chaos regimes, no crossover to the behavior $Lambda propto t^{-1}$ (which is indicative of regular motion) is observed. 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.
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Do nonlinear waves destroy Anderson localization? Computational and experimental studies yield subdiffusive nonequilibrium wave packet spreading. Chaotic dynamics and phase decoherence assumptions are used for explaining the data. We perform a quantitative analysis of the nonequilibrium chaos assumption, and compute the time dependence of main chaos indicators - Lyapunov exponents and deviation vector distributions. We find a slowing down of chaotic dynamics, which does not cross over into regular dynamics up to the largest observed time scales, still being fast enough to allow for a thermalization of the spreading wave packet. Strongly localized chaotic spots meander through the system as time evolves. Our findings confirm for the first time that nonequilibrium chaos and phase decoherence persist, fueling the prediction of a complete delocalization.
We probe the limits of nonlinear wave spreading in disordered chains which are known to localize linear waves. We particularly extend recent studies on the regimes of strong and weak chaos during subdiffusive spreading of wave packets [EPL {bf 91}, 30001 (2010)] and consider strong disorder, which favors Anderson localization. We probe the limit of infinite disorder strength and study Frohlich-Spencer-Wayne models. We find that the assumption of chaotic wave packet dynamics and its impact on spreading is in accord with all studied cases. Spreading appears to be asymptotic, without any observable slowing down. We also consider chains with spatially inhomogeneous nonlinearity which give further support to our findings and conclusions.
Chaotic systems exhibit rich quantum dynamical behaviors ranging from dynamical localization to normal diffusion to ballistic motion. Dynamical localization and normal diffusion simulate electron motion in an impure crystal with a vanishing and finite conductivity, i.e., an Anderson insulator and a metal, respectively. Ballistic motion simulates a perfect crystal with diverging conductivity, i.e., a supermetal. We analytically find and numerically confirm that, for a large class of chaotic systems, the metal-supermetal dynamics crossover occurs and is universal, determined only by the systems symmetry. Furthermore, we show that the universality of this dynamics crossover is identical to that of eigenfunction and spectral fluctuations described by the random matrix theory.
We investigate the behavior of the Generalized Alignment Index of order $k$ (GALI$_k$) for regular orbits of multidimensional Hamiltonian systems. The GALI$_k$ is an efficient chaos indicator, which asymptotically attains positive values for regular motion when $2leq k leq N$, with $N$ being the dimension of the torus on which the motion occurs. By considering several regular orbits in the neighborhood of two typical simple, stable periodic orbits of the Fermi-Pasta-Ulam-Tsingou (FPUT) $beta$ model for various values of the systems degrees of freedom, we show that the asymptotic GALI$_k$ values decrease when the indexs order $k$ increases and when the orbits energy approaches the periodic orbits destabilization energy where the stability island vanishes, while they increase when the considered regular orbit moves further away from the periodic one for a fixed energy. In addition, performing extensive numerical simulations we show that the indexs behavior does not depend on the choice of the initial deviation vectors needed for its evaluation.
We numerically investigate the characteristics of chaos evolution during wave packet spreading in two typical one-dimensional nonlinear disordered lattices: the Klein-Gordon system and the discrete nonlinear Schr{o}dinger equation model. Completing previous investigations cite{SGF13} we verify that chaotic dynamics is slowing down both for the so-called `weak and `strong chaos dynamical regimes encountered in these systems, without showing any signs of a crossover to regular dynamics. The value of the finite-time maximum Lyapunov exponent $Lambda$ decays in time $t$ as $Lambda propto t^{alpha_{Lambda}}$, with $alpha_{Lambda}$ being different from the $alpha_{Lambda}=-1$ value observed in cases of regular motion. In particular, $alpha_{Lambda}approx -0.25$ (weak chaos) and $alpha_{Lambda}approx -0.3$ (strong chaos) for both models, indicating the dynamical differences of the two regimes and the generality of the underlying chaotic mechanisms. The spatiotemporal evolution of the deviation vector associated with $Lambda$ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattices excited part.
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