Energy spreading, equipartition, and chaos in lattices with non-central forces
Arnold Ngapasare1, Georgios Theocharis2, Olivier Richoux2, Vassos Achilleos2, and Charalampos Skokos1,†
1 Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa; 2 Laboratoire d'Acoustique de l'Université du Mans(LAUM), UMR 6613, Institut d'Acoustique-Graduate School(IA-GS), CNRS, Le Mans Université, France
Abstract We numerically study a one-dimensional, nonlinear lattice model which in the linear limit is relevant to the study of bending (flexural) waves. In contrast with the classic one-dimensional mass-spring system, the linear dispersion relation of the considered model has different characteristics in the low frequency limit. By introducing disorder in the masses of the lattice particles, we investigate how different nonlinearities in the potential (cubic, quadratic, and their combination) lead to energy delocalization, equipartition, and chaotic dynamics. We excite the lattice using single site initial momentum excitations corresponding to a strongly localized linear mode and increase the initial energy of excitation. Beyond a certain energy threshold, when the cubic nonlinearity is present, the system is found to reach energy equipartition and total delocalization. On the other hand, when only the quartic nonlinearity is activated, the system remains localized and away from equipartition at least for the energies and evolution times considered here. However, for large enough energies for all types of nonlinearities we observe chaos. This chaotic behavior is combined with energy delocalization when cubic nonlinearities are present, while the appearance of only quadratic nonlinearity leads to energy localization. Our results reveal a rich dynamical behavior and show differences with the relevant Fermi-Pasta-Ulam-Tsingou model. Our findings pave the way for the study of models relevant to bending (flexural) waves in the presence of nonlinearity and disorder, anticipating different energy transport behaviors.
Fund: Ch. S. thanks the Université du Mans for its hospitality during his visits when part of this work was carried out. We also thank the Centre for High Performance Computing (https://www.chpc.ac.za) for providing computational resources for performing significant parts of this paper's computations. A. N. acknowledges funding from the University of Cape Town (University Research Council, URC) postdoctoral Fellowship grant and the Oppenheimer Memorial Trust (OMT) postdoctoral Fellowship grant.
Arnold Ngapasare, Georgios Theocharis, Olivier Richoux, Vassos Achilleos, and Charalampos Skokos Energy spreading, equipartition, and chaos in lattices with non-central forces 2022 Chin. Phys. B 31 020506
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