Generation of Localized Modes in an Electrical Lattice Using Subharmonic Driving

Abstract

We show experimentally and numerically that an intrinsic localized mode (ILM) can be stably produced (and experimentally observed) via subharmonic, spatially homogeneous driving in the context of a nonlinear electrical lattice. The precise nonlinear spatial response of the system has been seen to depend on the relative location in frequency between the driver frequency, ! d , and the bottom of the linear dispersion curve, ! 0 . If ! d =2 lies just below ! 0 , then a single ILM can be generated in a 32-node lattice, whereas, when ! d =2 lies within the dispersion band, a spatially extended waveform resembling a train of ILMs results. To our knowledge, and despite its apparently broad relevance, such an experimental observation of subharmonically driven ILMs has not been previously reported. DOI: 10.1103/PhysRevLett.108.084101 PACS numbers: 05.45.Yv, 63.20.Pw, 63.20.Ry It is well known that a damped nonlinear oscillator can respond at its intrinsic resonance frequency when it is driven at a multiple of that frequency. A direct example of such subharmonic driving is provided by the driven Van der Pol oscillator, where the ratio of response to driver frequency is exactly 1=3 [1,2]. Many other nonlinear oscillators exhibit similar subharmonic resonances (the Duffing oscillator being another extensively studied example). In fact, subharmonic response must be seen as a fairly generic property of nonlinear oscillators. Alternatively, a nonlinear oscillator with a parameter modulated at a particular frequency can also respond at a fraction of that frequency in what is called parametric excitation.What happens when such nonlinear oscillators are connected to one another in a regular lattice? In nonlinear lattices, an important generic phenomenon is the existence of self-trapped localized modes, known as intrinsic localized modes (ILMs) or discrete breathers. Such a mode represents an excitation which is (typically exponentially) spatially localized over a limited range of lattice nodes and decays to zero far from these, and it is temporally periodic. In this regard, it can be thought of as an analog of the solitons of continuous media. However, the discreteness of the lattice introduces interesting variations to the problem, including, for instance, the fact that ILMs may be dynamically stable in any dimension. Here, we blend these two broadly significant aspects of nonlinear systems by addressing the following question: can subharmonic or parametric excitations, which figure so prominently in isolated nonlinear oscillators, carry over to the lattice setting? That is, we examine whether ILMs can be generated and, especially, stabilized by subharmonic and/or parametric driving which is homogeneous in space. So far, this type of question seems to have been considered chiefly in the context of continuous media [10], or for parametric driving [11][12][13], and has been principally theoretical in nature. In this Letter, we demonstrate experimentally and corroborate through theoretical modeling and numerical computation, an...

Nonlinear localized modes in two-dimensional electrical lattices

“…As in the one-dimensional line [28], we again observe subharmonic breathers (experimentally and theoretically) in two dimensions. In Fig.…”

Instability dynamics and breather formation in a horizontally shaken pendulum chain

“…18(a) and 18(b), respectively, where we have plotted θ (t) for each pendulum over three forcing periods, for ω = 2.7 and N = 101. Such so-called subharmonic breathers have also been found in the context of electrical lattices recently [16]. Examining the Floquet spectrum for these families we find that the in-phase state is highly unstable (dashed line in Fig.…”

“…[10], as well as the electrical ones of Ref. [16]. On the other hand, understanding more systematically breather, as well as multibreather states and a potential tuning of their existence, as well as stability regimes would be of particular interest in inducing (and optimizing) energy localization in this system.…”

Effects of parity-time symmetry in nonlinear Klein-Gordon models and their stationary kinks

“…This is of interest in its own right, not only due to the fundamental mathematical and physical features that arise therein (radiation, internal modes, Peierls-Nabarro barriers, etc. [36,37]), but also because the discrete model arises in experimentally relevant mechanical and electrical systems [31,32,38,39], including PT -symmetric ones.…”