Variability in Fermi-Pasta-Ulam-Tsingou arrays can prevent recurrences

In 1955, Fermi, Pasta, Ulam, and Tsingou reported recurrence over time of energy between modes in a one-dimensional array of nonlinear oscillators. Subsequently, there have been myriad numerical experiments using homogenous FPUT arrays in the form of chains of ideal, nonlinearly coupled oscillators. However, inherent variations (e.g., due to manufacturing tolerance) introduce heterogeneity into the parameters of any physical system. We demonstrate that such tolerances degrade the observance of recurrences, often leading to complete loss in moderately-sized arrays. We numerically simulate heterogeneous FPUT systems to investigate the effects of tolerances on dynamics. Our results illustrate that tolerances in real nonlinear oscillator arrays may limit the applicability of results from numerical experiments on them to physical systems, unless appropriate heterogeneities are taken into account.

Figure 1

A 1D system of masses coupled via nonlinear springs in a similar manner to that used by Fermi, Pasta, Ulam, and Tsingou in their numerical experiments.

Figure 2

The FPUT recurrence phenomenon in a 64-particle system (1) with nonlinearity coefficient $\alpha =0.25$. We show the first four modes of the system. Observe that the energy moves between the modes, almost disappearing from the first mode before returning to the first mode some time later.

Figure 3

Simulations of a 1D FPUT-$\alpha$ array with $N=64$ oscillators and a nonlinearity coupling coefficient of $\alpha =0.25$ showing examples of recurrence at a tolerance of $\pm 1\%$. These are samples of the range of responses when we use a set of tolerance values that we generate randomly from a Gaussian distribution. Panel (a) is representative of the response in more than $85\%$ of the cases with $N=64$, $\alpha =0.25$, and a tolerance of $\pm 1\%$. One can observe that recurrence is present, but it is a bit different from what occurs in the ideal scenario. In some cases, recurrence is weaker [as in panel (b)] or the energy fails to return to the first mode [as in panel (c)].

Figure 4

Simulation of a 1D FPUT-$\alpha$ array with symmetric tolerances, 64 oscillators, and a coupling coefficient of $\alpha =0.25$ at a tolerance of $\pm 0.1\%$. Comparing this to Fig. 2, we see a slight variation from the response of an ideal system.

Figure 5

The effect of tolerance on different parts of a 1D FPUT-$\alpha$ array with a coupling coefficient of $\alpha =0.25$ and $N=64$ oscillators. Panels (a)–(c) show examples with tolerance in both the linear and nonlinear terms, panels (d)–(f) show examples with tolerance in only the linear terms, and panels (g)–(i) show examples with tolerance in only the nonlinear terms. Observe that adding tolerance to only the linear terms has a comparable effect on recurrence to adding tolerance to only the nonlinear terms. Tolerance in both linear and nonlinear terms: (a) $\pm 1\%$, (b) $\pm 5\%$, and (c) $\pm 10\%$. Tolerance in the linear terms: (d) $\pm 1\%$, (e) $\pm 5\%$, and (e) $\pm 10\%$. Tolerance in the nonlinear terms: (g) $\pm 1\%$, (h) $\pm 5\%$, and (i) $\pm 10\%$.

Figure 6

Simulations of 1D FPUT-$\alpha$ arrays with different numbers of oscillators ($N$) at a tolerance of $\pm 5\%$ with a coupling coefficient of $\alpha =0.25$. (a) $N=8$, (b) $N=16$, (c) $N=32$, (d) $N=64$, and (e) $N=128$. [Panel (d) appeared previously in Fig. 5.] This figure illustrates that, for a given tolerance, arrays with fewer oscillators are more likely to exhibit recurrence (which does not occur if there are too many oscillators).

Figure 7

Simulation of a 1D FPUT-$\alpha$ array with asymmetric tolerances, 64 oscillators, and a coupling coefficient of $\alpha =0.25$ at a tolerance of $\pm 0.1\%$. Comparing this simulation to that in Fig. 4, we observe that this asymmetric system has a comparable deviation from ideal recurrence (i.e., when there is no tolerance) to the symmetric system.

Figure 8

The effect of tolerance on different parts of an asymmetric 1D FPUT-$\alpha$ array with $N=64$ oscillators and a coupling coefficient of $\alpha =0.25$. Comparing this simulation to that in Fig. 5, we see that incorporating tolerance in only the nonlinear terms has a much smaller effect on the qualitative dynamics than is the case in the symmetric system. Tolerance in both linear and nonlinear terms: (a) $\pm 1\%$, (b) $\pm 5\%$, and (c) $\pm 10\%$. Tolerance in the linear terms: (d) $\pm 1\%$, (e) $\pm 5\%$, and (f) $\pm 10\%$. Tolerance in the nonlinear terms: (g) $\pm 1\%$, (h) $\pm 5\%$, and (i) $\pm 10\%$.

Figure 9

Simulations of 1D FPUT-$\alpha$ arrays with different numbers of oscillators ($N$) at a tolerance of $\pm 5\%$ with a coupling coefficient of $\alpha =0.25$. (a) $N=8$, (b) $N=16$, (c) $N=32$, (d) $N=64$, and (e) $N=128$. [Panel (d) appeared previously in Fig. 8.]

Figure 10

The effect of tolerance on different parts of an asymmetric 1D FPUT-$\alpha$ array with equations of motion (A1), $N=64$ oscillators, a coupling coefficient of $\alpha =0.25$, and a tolerance of $\pm 5\%$. Panel (a) shows a simulation with tolerance in both the linear and nonlinear terms, panel (b) shows a simulation with tolerance in only the linear terms, and panel (c) shows a simulation with tolerance in only the nonlinear terms.

Figure 11

The effect of tolerance on different parts of an asymmetric 1D FPUT-$\alpha$ array with equations of motion (A2), $N=64$ oscillators, a coupling coefficient of $\alpha =0.25$, and a tolerance of $\pm 5\%$. Panel (a) shows a simulation with tolerance in both the linear and nonlinear terms, panel (b) shows a simulation with tolerance in only the linear terms, and panel (c) shows a simulation with tolerance in only the nonlinear terms.

Figure 12

Output from the time-reversed simulation of Fig. 3, showing that we obtain values close to the initial condition (which consists of an excitation of the system's first mode).