Observations for practical Fermi–Pasta–Ulam–Tsingou systems

Nonlinear dynamics is a field that has been widely studied by mathematicians and scientists. However, engineers are yet to exploit the full potential of nonlinearity in coupled systems despite the majority of real physical systems being inherently nonlinear. Although a vast array of results has been generated from simulations, only a small number of physical systems have been investigated. In a 1955 report, Fermi, Pasta, Ulam, and Tsingou reported recurrence of energy between modes in a simple mass-spring system containing a nonlinear coupling term. Extensive numerical simulations of the FPUT system have followed this discovery. However, minimal evidence of FPUT recurrence has been observed in physical systems. This may be due to the effect of ‘real-world’ imperfections on the ideal systems used in the simulations. Variations between system components due to manufacturing tolerances and energy losses through damping are present in most physical systems. By understanding the effects of these imperfections, it may be possible to apply the results of the simulations to physical systems. The effects of tolerance and damping on recurrence were studied using a series of simulations. The results demonstrate that such imperfections degrade the observation of recurrence, often leading to complete loss in moderately sized arrays. These simulations enabled the identification of a series of conditions under which a practical system might display recurrence. Tight tolerances, low damping, small array size and moderately weak coupling strengths are required. An electronic amplifier circuit was designed to meet these constraints. Preliminary testing of this electronic system has provided early verification of the simulation results. FPUT recurrence was observed a system with four elements. These results suggest that without considering tolerances in real systems, the applicability of the results from simulations to physical systems may be limited.