Nonlinear Coupled Oscillators. II. Comparison of Theory with Computer Solutions

• Published in: Journal of mathematical physics 1963-05, Vol.4 (5), p.686-700
• Main Author: Jackson, E. Atlee
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.1704007

A study is made of a particular system of coupled oscillators to determine how the amount of energy exchange and the recurrence time depend on the parameters of the system, and the accuracy with which these are predicted by the perturbation theory developed in part I of this series. The system consists of N + 1 particles, connected by springs which have a quadratic force term, the strength of which, λ i , can vary between different particles (``defects''). The equations of motion for the perfect chain (λ i = λ) were solved on a computer for the cases N = 4, λ = ¼, ½, ¾; and N = 9, λ = ½; and supplemented by the previous calculations of Fermi, Pasta, and Ulam, in which N = 32, λ = ¼; 1. The ``energy'' in the linear modes, E k (t)= 1 2 (ȧ k 2 +ω k 2 a k 2 ) are determined from these computations and compared with the perturbation theory in the first paper of this series. As was found by Fermi, Pasta, and Ulam, when only E 1 is initially excited, then only the first few modes acquire an appreciable amount of energy, after which nearly all the energy returns to the first mode in a time τλ. The second‐order perturbation theory is found to give an accurate estimate (within 15%) of both τλ and the amount of energy exchange for all the above cases, except N = 32, λ = 1, which requires higher‐order analysis. It is shown that the nonergodic behavior of this system does not result simply from the incommensurability of the uncoupled frequencies {ω k }, but also from the particular form of mode interaction and the initial conditions used in all calculations, both of which affect the coupled frequency spectrum {Ω k }. To alter the mode interaction a preliminary study is made of ``imperfect'' chains (variable λ i ), which directly couples low and high modes. It is found that either a large, or else systematic, variation of λ i is necessary to appreciably alter the nonergodic behavior of this system. A theoretical examination of the dependence of the coupled frequencies {Ω k } on the initial conditions shows that the ergodic behavior will, in general, be strongly dependent upon the initial conditions.