Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains

TL;DR

This paper explores how energy localization in nonlinear oscillatory chains is governed by limiting phase trajectories, extending the beating concept to finite chains, and highlighting the role of effective particles and resonance interactions.

Contribution

It introduces the concept of Limiting Phase Trajectories and effective particles to describe energy transfer and localization in finite nonlinear oscillatory chains.

Findings

Energy transfer requires excitation levels twice the instability threshold.

Limiting Phase Trajectories serve as an alternative to Nonlinear Normal Modes.

Resonance interactions between zone boundary and nearby modes facilitate energy exchange.

Abstract

We demonstrate that the modulation instability of the zone boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone boundary mode's instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of 'effective particles' and Limiting Phase Trajectories. The 'effective particles' appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the Limiting Phase Trajectories corresponding to the most intensive energy exchange between 'effective particles' can be considered as an alternative to Nonlinear Normal Modes, which describe the stationary process.