  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, utilising 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.
