  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.
