Chaos and Anderson Localisation in Disordered Classical Chains: Hertzian vs FPUT models

We numerically investigate the dynamics of an one-dimensional disordered lattice using the Hertzian model, describing a granular chain, and the $\alpha+\beta $ Fermi-Pasta-Ulam-Tsingou model (FPUT). The most profound difference between the two systems is the discontinuous nonlinearity of the granular chain appearing whenever neighboring particles are detached. We therefore sought to unravel the role of these discontinuities in the destruction of Anderson localization and their influence on the system's chaotic dynamics. Our results show that both models exhibit an energy range where localization coexists with chaos. However, the discontinuous nonlinearity is found to be capable of triggering energy spreading of initially localized modes, at lower energies than the FPUT model. A transition from Anderson localization to chaotic dynamics and energy equipartition is found for the granular chain and is associated with the"propagation" of the discontinuous nonlinearity in the chain. On the contrary, the FPUT chain exhibits an alternate behavior between localized and delocalized chaotic behavior which is strongly dependent on the initial energy of excitation.