We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a “graphene-type” force law and one inspired by Hollomon’s law describing “work-hardening” effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of nonanalytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi–Pasta–Ulam–Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles [Formula: see text] and the total energy [Formula: see text] are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large [Formula: see text], it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.

中文翻译：

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具有非解析势的一维哈密顿晶格的稳定性性质

我们研究了两个 1 维 (1D) 哈密顿晶格的局部和全局动力学，其粒子间力来自非解析势。特别是，我们研究了由“石墨烯型”力定律控制的模型的动力学，以及受描述某些弹性材料中“加工硬化”效应的 Hollomon 定律启发的模型的动力学。我们的主要目的是表明，尽管存在与解析情况的相似之处，但非解析势的一些局部和全局稳定性属性与具有多项式相互作用的系统中遇到的那些非常不同，例如一维 Fermi-Pasta-Ulam 的情况–Tsingou (FPUT) 晶格。我们的方法是研究表示相应线性系统的正常模式的延续的简单周期轨道附近的运动，作为粒子的数量[公式：见正文]和总能量[公式：见正文]增加。我们发现，石墨烯型模型非常稳定，可以逃脱预期击穿的能级，而 Hollomon 晶格从不断裂，但在低能量下不稳定，只有在谐波力占主导地位的能量下才能达到稳定性。我们建议，由于我们的结果适用于大 [公式：见文本]，因此研究 1D 晶格变为字符串的连续极限中的类似现象会很有趣。