Starosvetsky Y, Ben-Meir Y
Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):062919. doi: 10.1103/PhysRevE.87.062919. Epub 2013 Jun 26.
In the present paper we study the mechanism that leads to the formation of regular patterns of energy localization and complete recurrent energy transport in the homogeneous systems of anharmonic oscillators and oscillatory chains subjected to a state of sonic vacuum. The basic model under investigation comprises a system of purely anharmonic oscillators as well as oscillatory chains given to a localized excitation where the initial energy is imported to one of the oscillators or oscillatory chains. The results of numerical simulations reveal the existence of a strong classical beating phenomenon, characterized by complete, recurrent, resonant energy exchanges between the oscillators and oscillatory chain and this in the state of sonic vacuum where no regular resonant frequencies can be defined. In this study we show that formation of the recurrent energy exchanges in this highly degenerate model is strictly stipulated by the system parameters. Thus, for instance, choosing the parameter of coupling below a certain threshold leads to significant energy localization on one of the oscillators or oscillatory chains. However, increasing the strength of coupling above the threshold leads to the formation of a strong beating response. The analytical study pursued in this paper predicts the origin of formation of a strong beating phenomenon and provides the necessary conditions on the system parameter for its excitation. Moreover, careful analysis of the beating phenomenon reveals the qualitatively different global bifurcation undergone by this type of highly nonstationary regime. The theoretical study is further extended to the system of coupled purely anharmonic lattices. Thus we show analytically and numerically that excitation of some particular solutions (e.g., spatially periodic standing waves and standing breathers) on one of the lattices results in the formation of similar patterns of energy (wave) localization as well as the regime of complete recurrent interchain energy transport. In particular we demonstrate that the formation of these regimes is solely affected by a particular choice of system parameters. The results of the analytical study are found to be in very good agreement with those of numerical simulations.
在本文中，我们研究了在处于声真空状态的非简谐振荡器和振荡链的均匀系统中，导致能量局域化的规则模式形成以及完全递归能量传输的机制。所研究的基本模型包括一个纯非简谐振荡器系统以及给定局部激发的振荡链，其中初始能量被输入到其中一个振荡器或振荡链中。数值模拟结果揭示了一种强烈的经典拍频现象的存在，其特征是振荡器和振荡链之间存在完全、递归、共振的能量交换，而且这是在无法定义规则共振频率的声真空状态下发生的。在本研究中，我们表明在这个高度简并的模型中，递归能量交换的形成严格由系统参数规定。例如，将耦合参数选择低于某个阈值会导致能量显著局域在其中一个振荡器或振荡链上。然而，将耦合强度增加到阈值以上会导致形成强烈的拍频响应。本文进行的分析研究预测了强烈拍频现象形成的起源，并为其激发提供了系统参数的必要条件。此外，对拍频现象的仔细分析揭示了这种高度非平稳状态所经历的定性不同的全局分岔。理论研究进一步扩展到耦合纯非简谐晶格系统。因此，我们通过解析和数值方法表明，在其中一个晶格上激发某些特定解（例如空间周期性驻波和驻波呼吸子）会导致形成类似的能量（波）局域模式以及完全递归的链间能量传输状态。特别是，我们证明了这些状态的形成仅受系统参数的特定选择影响。分析研究结果与数值模拟结果非常吻合。
