Spreading of wave packets in disordered systems with tunable nonlinearity

We study the spreading of single-site excitations in one-dimensional disordered Klein-Gordon chains with tunable nonlinearity $|u_{l}|^{\sigma} u_{l}$ for different values of $\sigma$. We perform extensive numerical simulations where wave packets are evolved a) without and, b) with dephasing in normal mode space. Subdiffusive spreading is observed with the second moment of wave packets growing as $t^{\alpha}$. The dependence of the numerically computed exponent $\alpha$ on $\sigma$ is in very good agreement with our theoretical predictions both for the evolution of the wave packet with and without dephasing (for $\sigma \geq 2$ in the latter case). We discuss evidence of the existence of a regime of strong chaos, and observe destruction of Anderson localization in the packet tails for small values of $\sigma$.Comment: 9 pages, 7 figure

Polydispersed Granular Chains: From Long-lived Chaotic Anderson-like Localization to Energy Equipartition

We investigate the dynamics of highly polydispersed finite granular chains. From the spatio-spectral properties of small vibrations, we identify which particular single-particle displacements lead to energy localization. Then, we address a fundamental question: Do granular nonlinearities lead to chaotic dynamics and if so, does chaos destroy this energy localization? Our numerical simulations show that for moderate nonlinearities, although the overall system behaves chaotically, it can exhibit long lasting energy localization for particular single particle excitations. On the other hand, for sufficiently strong nonlinearities, connected with contact breaking, the granular chain reaches energy equipartition and an equilibrium chaotic state, independent of the initial position excitation

Global dynamics of coupled standard maps

Understanding the dynamics of multi--dimensional conservative dynamical systems (Hamiltonian flows or symplectic maps) is a fundamental issue of non-linear science. The Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems \cite{sk:6}, is an ideal tool for this purpose. In the present paper we make a first step towards the dynamical study of multi--dimensional maps, by obtaining some interesting results for a 4--dimensional (4D) symplectic map consisting of N=2 coupled standard maps \cite{Kan:1}. In particular, using the new GALI$_3$ and GALI$_4$ indices, we compute the percentages of regular and chaotic motion of the map equally reliably but much faster than previously used indices, like GALI$_2$ (known in the literature as SALI).Comment: 4 pages, 3 figures, to appear in the proceedings of the international conference "Chaos in Astronomy", Athens, Greece (poster contribution

Chaoticity without thermalisation in disordered lattices

We study chaoticity and thermalization in Bose-Einstein condensates in disordered lattices, described by the discrete nonlinear Schr\"odinger equation (DNLS). A symplectic integration method allows us to accurately obtain both the full phase space trajectories and their maximum Lyapunov exponents (mLEs), which characterize their chaoticity. We find that disorder destroys ergodicity by breaking up phase space into subsystems that are effectively disjoint on experimentally relevant timescales, even though energetically, classical localisation cannot occur. This leads us to conclude that the mLE is a very poor ergodicity indicator, since it is not sensitive to the trajectory being confined to a subregion of phase space. The eventual thermalization of a BEC in a disordered lattice cannot be predicted based only on the chaoticity of its phase space trajectory

Detecting order and chaos in Hamiltonian systems by the SALI method

We use the Smaller Alignment Index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behavior of the SALI for the two cases: the index fluctuates around non--zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behavior for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents $\sigma_1$, $\sigma_2$ i.e. $SALI \propto e^{-(\sigma_1-\sigma_2)t}$. Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of 2 and 3 degrees of freedom.Comment: 18 pages, 10 figures, accepted for publication in J. Phys.

Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices

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 lattice's excited part.Comment: 11 pages, 10 figure