Stability Properties of 1-Dimensional Hamiltonian Lattices with Nonanalytic Potentials

Abstract

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.

Nonlinear topological edge states: From dynamic delocalization to thermalization

“…In general, the stability properties of autonomous Hamiltonian system's are well described in the framework of phase-space dynamics. Indeed, we expect stable nonlinear topological edge states to belong to regular "islands" surrounded by chaotic "sea" [72,77]. In the prospective of Secs.…”

“…which tells us how far we are from the topological edge state, i.e., the radius of the regular island [77]. In Eq.…”

Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization

“…In general, the stability properties of autonomous Hamiltonian system's are well described in the framework of phase space dynamics. Indeed, we expect stable nonlinear topological edge states to belong to regular 'islands' surrounded by chaotic 'sea' [71,76]. In the prospective of Secs.…”

“…which tells us how far we are from the topological edge state, i.e., the radius of the regular island [76]. In Eq.…”