Energy-recurrence breakdown and chaos in disordered Fermi–Pasta–Ulam–Tsingou lattices

In this paper, we consider the classic Fermi–Pasta–Ulam–Tsingou system as a model of interacting particles connected by harmonic springs with a quadratic nonlinear term (first system) and a set of second-order ordinary differential equations with variability (second system) that resembles Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system. In the absence of variability, the second system becomes Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system (first system). Variability is introduced to Hamilton’s equations of motion of the Fermi–Pasta–Ulam–Tsingou system to take into account inherent variations (for example, due to manufacturing processes), giving rise to heterogeneity in its parameters. We demonstrate that a percentage of variability smaller than a threshold can break the well-known energy recurrence phenomenon and induce localization in the energy normal-mode space. However, percentage of variability larger than the threshold may make the trajectories of the second system blow up in finite time. Using a multiple-scale expansion, we derive analytically a two normal-mode approximation that explains the mechanism for energy localization and blow up in the second system. We also investigate the chaotic behavior of the two systems as the percentage of variability is increased, utilizing the maximum Lyapunov exponent and Smaller Alignment Index. Our analysis shows that when there is almost energy localization in the second system, it is more probable to observe chaos, as the number of particles increases.