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 Schrodinger equation model. Completing previous investigations [Ch. Skokos et al., Phys. Rev. Lett. 111, 064101 ( 2013)], we verify that chaotic dynamics is slowing down for both 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 proportional to t(alpha Lambda) with alpha(Lambda) being different from the alpha(Lambda) = -1 value observed in cases of regular motion. In particular, alpha(Lambda) approximate to -0.25 (weak chaos) and alpha(Lambda) approximate to -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.