Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices

Abstract

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 Schrödinger equation model. Completing previous investigations [38] 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 Λ decays in time t as Λ ∝ t α Λ , with αΛ being different from the αΛ = −1 value observed in cases of regular motion. In particular, αΛ ≈ −0.25 (weak chaos) and αΛ ≈ −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 Λ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattice's excited part.

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

“…2. Again the results obtained by these integrators are practically the same for both sets of initial conditions, reproducing the tendency of the finite time mLE to asymptotically decrease according to the power law X 1 (t) ∝ t α L with α L ≈ −0.3 (case I 1D ) and α L ≈ −0.25 (case II 1D ), in accordance to the results of [32,33].…”

“…From the results of this figure we see that X 1 exhibits a tendency to decrease following a completely different decay from the X 1 ∝ t −1 power law observed for regular motion. This behavior was also observed for the 2D DKG model [56], as well as for the 1D DKG and DDNLS systems in [32,33] where a power law X 1 (t) ∝ t α L with α L ≈ −0.25 and α L ≈ −0.3 for, respectively, the weak and strong chaos dynamical regimes was established. Further investigations of the behavior of the finite mLE in 2D disordered systems are required in order to determine a potentially global behavior of X 1 , since here and in [56] only some isolated cases were discussed.…”

“…We note that cases I 1D and II 1D have been studied in [33] and respectively correspond to the so-called 'strong chaos' and 'weak chaos' dynamical regimes of this model. As initial normalized deviation vector we use a vector having non-zero coordinates only at the central site of the lattice, while its remaining elements are set to zero.…”

“…where j = N j=1 jζ j is the position of the center of the distribution [22,24,26,29,33,49,50]. We also check how accurately the values of the system's two conserved quantities, i.e.…”

Contribution of individual degrees of freedom to Lyapunov vectors in many-body systems

“…to estimate its individual participation to the divergence rate between nearby trajectories in 2N -dimensional phase space. This weight (normalised to unity) can be interpreted as the sensitivity level of the region where particle k is in the kinetic space, which leads to the operative notion of Lyapunov modes [8-10, 21, 27, 40] and also sheds light on the role of deterministic chaos and dephasing in the effect of nonlinear dynamics on Anderson localization [36,38].…”