  We study instabilities and relaxation to equilibrium in a long-range
extension of the Fermi-Pasta-Ulam-Tsingou (FPU) oscillator chain by exciting
initially the lowest Fourier mode. Localization in mode space is stronger for
the long-range FPU model. This allows us to uncover the sporadic nature of
instabilities, i.e., by varying initially the excitation amplitude of the
lowest mode, which is the control parameter, instabilities occur in narrow
amplitude intervals. Only for sufficiently large values of the amplitude, the
system enters a permanently unstable regime. These findings also clarify the
long-standing problem of the relaxation to equilibrium in the short-range FPU
model. Because of the weaker localization in mode space of this latter model,
the transfer of energy is retarded and relaxation occurs on a much longer
time-scale.
