Instability dynamics and breather formation in a horizontally shaken pendulum chain

Abstract

Inspired by the experimental results of Cuevas et al. [Phys. Rev. Lett. 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 a 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 multifrequency 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.

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

“…In general, we have observed similar behavior to that found in [21], where the same system was studied in a slightly different range of parameters. Examining the response of the system to different frequencies and amplitudes of the driving force, we obtain the resonance curves shown in Fig.…”

“…In our system, in contrast, we have been able to identify such a mode experimentally for a range of frequencies, and its existence and dynamical stability have been corroborated by numerical computations. It should also be noted that, as indicated also in [21], the precise stability details of such a subharmonic breather depend strongly on the number of pendula in the chain, with a smaller N favoring more robust configurations. The mode profile corresponding to a frequency close to 2 Hz is mapped out in Fig.…”

“…This damped-driven system was previously illustrated to feature prototypical discrete breather waveforms. Recent explorations [21] have suggested the possibility of subharmonic breather structures, which are unstable for larger chains but potentially stable for smaller lattices. In the present work, we have confirmed this expectation via a combination of theoretical modeling (and where possible analysis), numerical computation and experimental realization.…”

“…[18]) may be used to excite discrete breather formation in an electrical lattice [19]. More recently, this idea has been examined further in the context of a horizontally shaken pendulum (which has long been known to display a variety of subharmonic resonances [20]), and the possibility of mixed-frequency breathers was identified in a pendulum chain [21]. These breathers exhibit the remarkable response that while energy is localized on a few pendula responding at a sub-harmonic of the driving force, the pendula in the tails of the breather are oscillating with the driving frequency.…”

Nonlinear coherent structures in granular crystals

“…9[b]) in a way that is reminiscent of the foldover event of driven oscillators [135]. See [195] for a recent study that extends this idea to granular chains. Such nonlinear bending can be useful for applications such as energy harvesting [35].…”