  Implementing the Generalized Alignment Index (GALI) method of chaos detection
we investigate the dynamical behavior of the nonlinear disordered Klein-Gordon
lattice chain in one spatial dimension. By performing extensive numerical
simulations of single site and single mode initial excitations, for several
disordered realizations and different disorder strengths, we determine the
probability to observe chaotic behavior as the system is approaching its linear
limit, i.e. when its total energy, which plays the role of the system's
nonlinearity strength, decreases. We find that the percentage of chaotic cases
diminishes as the energy decreases leading to exclusively regular motion on
multidimensional tori. We also discriminate between localized and spreading
chaos, with the former dominating the dynamics for lower energy values. In
addition, our results show that single mode excitations lead to more chaotic
behaviors for larger energies compared to single site excitations. Furthermore,
we demonstrate how the GALI method can be efficiently used to determine a
characteristic chaoticity time scale for the system when strong enough
nonlinearites lead to energy delocalization in both the so-called `weak' and
`strong chaos' spreading regimes.
