Instability dynamics and breather formation in a horizontally shaken pendulum chain

TL;DR

This paper investigates the nonlinear dynamics of a horizontally shaken pendulum chain, revealing conditions for instability, breather formation, and stable localized oscillations, supported by theoretical analysis and a robust breather generation scheme.

Contribution

It provides a comprehensive theoretical analysis of instability and breather formation in a driven pendulum chain, including new insights into multi-frequency breathers and generation methods.

Findings

Critical chain length and frequency for modulational instability identified

Connection between breather solutions and single pendulum dynamics established

Robust scheme for generating on-site and off-site breathers demonstrated

Abstract

Inspired by the experimental results of Cuevas et al. (Physical Review Letters 102, 224101 (2009)), we consider theoretically the behavior of a chain of planar rigid pendulums suspended in a uniform gravitational field and subjected to a horizontal periodic driving force applied to the pendulum pivots. We characterize the motion of a single pendulum, finding bistability near the fundamental resonance, and near the period-3 subharmonic resonance. We examine the development of modulational instability in a driven pendulum chain and find both a critical chain length and critical frequency for the appearance of the instability. We study the breather solutions and show their connection to the single pendulum dynamics, and extend our analysis to consider multi-frequency breathers connected to the period-3 periodic solution, showing also the possibility of stability in these breather states. Finally we examine the problem of breather generation and demonstrate a robust scheme for generation of on-site and off-site breathers.