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 $\mathrm{\Lambda }$ decays in time $t$ as $\Lambda \propto t^{\alpha_{\Lambda}}$,with$\alpha_{\Lambda}$beingdifferentfromthe$\alpha_{\Lambda}=-1$valueobservedincasesofregularmotion\mathrm{.}Inparticular,$\alpha_{\Lambda}\approx -0.25$(weakchaos)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 $\mathrm{\Lambda }$ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattice’s excited part.