Intrinsic localized modes in parametrically driven arrays of nonlinear resonators

Kenig, Eyal; Malomed, Boris A.; Cross, M. C.; Lifshitz, Ron

doi:10.1103/physreve.80.046202

Cited by 85 publications

(74 citation statements)

References 72 publications

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“…Such arrays have already exhibited interesting nonlinear dynamics, ranging from the formation of extended patterns [8,38], as one commonly observes in analogous continuous systems such as Faraday waves, to that of intrinsically localized modes [39,[58][59][60]. Thus, nanomechanical resonator arrays are perfect for testing dynamical theories of discrete nonlinear systems with many degrees of freedom.…”

Section: Why Study Nonlinear Nems and Mems?mentioning

confidence: 99%

“…This extension includes other experimentally relevant questions, such as the response of the system of coupled resonators to time dependent sweeps of the control parameters, rather than quasistatic sweeps like the ones we have been discussing here. Kenig et al [39] have also studied the formation and dynamics of intrinsically-localized modes, or solitons, in the array equations of LC. To this end, they derived a different amplitude equation, which takes the form of a parametrically-driven damped nonlinear Shrödinger equation, also known as a forced complex Ginzburg-Landau equation.…”

Section: Parametric Excitation Of Arrays Of Coupled Duffing Resonatorsmentioning

confidence: 99%

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Nonlinear Dynamics of Nanomechanical Resonators

Lifshitz

Cross

2010

Nonlinear Dynamics of Nanosystems

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Section: Why Study Nonlinear Nems and Mems?mentioning

confidence: 99%

Section: Parametric Excitation Of Arrays Of Coupled Duffing Resonatorsmentioning

confidence: 99%

Nonlinear Dynamics of Nanomechanical Resonators

Lifshitz

Cross

2010

Nonlinear Dynamics of Nanosystems

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“…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, and, when possible, infusing analytical insights, that ILMs can indeed be generated and stabilized via subharmonic forcing.…”

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confidence: 99%

Generation of Localized Modes in an Electrical Lattice Using Subharmonic Driving

et al. 2012

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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...

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“…The sine-Gordon model and its discrete analog are ubiquitous models in mathematical physics with a wide range of applications extending from chains of coupled pendulums [1] and Josephson junction arrays [2] to gravitational and high-energy physics models [3,4]. For instance, Ikeda et al investigated the behavior of intrinsic localized modes (ILMs) for an array of coupled pendulums subjected to horizontal [5] and verical [6] sinusoidal excitation.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamics of a 2d Array of Coupled Pendulums Under Parametric Excitation

Jallouli¹,

Kacem²,

Bouhaddi³

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. In this paper, we investigate the frequency response of a 2D array of coupled pendulums under parametric excitation. We developed a computational model while considering the main source of nonlinearities. The nonlinear equations of motion are derived and solved using the Harmonic Balance Method (HBM) coupled with the Asymptotic Numerical Method (ANM). Numerical simulation are performed in the case of 3X3 array to investigate the effect of changing the coupling coefficients in the 2D direction and adding an imperfection on the collective dynamics of the array. 3057

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Intrinsic localized modes in parametrically driven arrays of nonlinear resonators

Cited by 85 publications

References 72 publications

Nonlinear Dynamics of Nanomechanical Resonators

Nonlinear Dynamics of Nanomechanical Resonators

Generation of Localized Modes in an Electrical Lattice Using Subharmonic Driving

Nonlinear Dynamics of a 2d Array of Coupled Pendulums Under Parametric Excitation

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