Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance

Manevitch, Leonid I.; Kovaleva, Agnessa; Shepelev, Denis S.

doi:10.1016/j.physd.2010.08.001

Cited by 55 publications

(42 citation statements)

References 15 publications

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“…2. It is seen that in the first half-cycle of oscillations, i.e., in the interval 0    T 0 , the amplitude a 0 () is close to the Limiting Phase Trajectory (LPT) of a moderately nonlinear system [22], but then a 0 () turns into relatively small oscillations near a monotonically increasing quasi-steady state a ̅ 0 (). The phase plot in the plane (, a 0 ) represents a spiral beginning at the point (-/2, 0) and converging to an upward center (0, a ̅ 0 ()) with a slowly increasing value of a ̅ 0 ().…”

Section: Quasi-steady States and Fast Fluctuations In Coupled Oscillamentioning

confidence: 99%

“…We recall that zero initial conditions define the so-called Limiting Phase Trajectories (LPTs) corresponding to motion with maximum possible energy transfer from a source of energy to a receiver [19,20]. Properties of the LPT in a nonlinear oscillator as well as transitions between different branches of the LPT employed in this work are determined by the forcing and nonlinearity parameters together with the non-zero initial detuning κ 1 [18,[22][23][24]. The singular case κ 1 = 0 is beyond the consideration.…”

Section: Main Equationsmentioning

confidence: 99%

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Response enhancement in an oscillator chain

Kovaleva

2016

Communications in Nonlinear Science and Numerical Simulation

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Section: Quasi-steady States and Fast Fluctuations In Coupled Oscillamentioning

confidence: 99%

Section: Main Equationsmentioning

confidence: 99%

Response enhancement in an oscillator chain

Kovaleva

2016

Communications in Nonlinear Science and Numerical Simulation

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“…В работе используется подход, основанный на концепции предельных фазовых траекторий (ПФТ), описывающих максимально интенсивный (при данных условиях) энер-гообмен между слабо взаимодействующими осцилляторами или кластерами осцилляторов (эффективными частицами). Эта концепция была разработана в ряде работ [3][4][5] и в настоя-щее время широко используется для исследования нестационарных резонансных процессов в классических и квантовых нелинейных системах [6][7][8][9][10][11]. Отметим, что важнейшая черта эффективной энергетической ловушки -ее принадлежность к классу систем, функцио-нирующих в условиях акустического вакуума.…”

Section: Introductionunclassified

Oscillatory chain with elastic supports and bending stiffness under conditions close to acoustic vacuum

Koroleva¹,

Manevich²

2016

Nelin. Dinam.

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Осцилляторная цепь c изгибной жесткостью на упругой подложке в условиях, близких к акустическому вакууму И. П. Королева (Кикоть), Л. И. МаневичПредставлены результаты аналитического и численного исследования нестационарной плоской динамики струны с равномерно распределенными дискретными массами при от-сутствии предварительного натяжения и с учетом относительно малой изгибной жесткости. Каждая масса испытывает также действие упругой подложки с нелинеаризуемой в усло-виях плоского движения характеристикой, которая тоже представляет собой струну без предварительного натяжения. Наиболее важный предельный случай, соответствующий низ-коэнергетическим поперечным возбуждениям, рассматривается с учетом геометрической нелинейности. Поскольку такие возбуждения описываются приближенными уравнениями, в которых наиболее существенный вклад вносят кубические упругие силы, колебания про-исходят фактически в условиях, близких к акустическому вакууму (термин «aкустический вакуум» означает, что рассматриваемая система не имеет не зависящих от aмплитуд ко-лебаний динамических характеристик, таких как собственные частоты и скорость звука). Получено адекватное аналитическое описание резонансных существенно нестационарных процессов в рассматриваемой системе, соответствующих интенсивному энергообмену меж-ду ее частями (кластерами) в области низких частот. Сформулированы условия локализа-ции энергии на одном из кластеров. Полученные аналитические результаты подтверждены данными компьютерного моделирования. Отмечено, что рассматриваемая система может использоваться как энергетическая ловушка повышенной эффективности.Ключевые слова: нелинейная динамика, нелинейная нормальная мода, предельная фазовая траектория, энергообмен, локализация

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“…Therefore only isolated and predominantly numerical results were obtained in this field [11,12]. The recently developed concept of Limiting Phase Trajectories (LPTs) allowed for a systematic approach to description of non-stationary resonance regimes [13][14][15][16][17][18], including coupled pendula dynamics [19]. This concept introduces a fundamental non-stationary process of new type which corresponds to maximum possible energy exchange between the oscillators (in particular, pendula) or clusters of oscillators.…”

mentioning

confidence: 99%

“…However, due to the restriction 0 θ π/2, they become saw-tooth type functions. The analytical solution of the problem in terms of non-smooth functions can be obtained after change of temporal variable through the procedure proposed in [22,23] and used for the study of non-stationary resonance processes in [13,15,16]. As for phase trajectories located inside the separatrix, they correspond to localized LPTs and can be easily found after linearization of the second order equation for LPTs in the vicinity of θ =θ = 0.…”

mentioning

confidence: 99%

Non-stationary resonance dynamics of weakly coupled pendula

Manevitch

Romeo

2015

EPL

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In this paper we fill the gap in understanding the non-stationary resonance dynamics of the weakly coupled pendula model, having significant applications in numerous fields of physics such as superconducting Josephson junctions, Bose-Einstein condensates, DNA, etc.. While common knowledge of the problem is based on two alternative limiting asymptotics, namely the quasi-linear approach and the approximation of independent pendula, we present a unified description in the framework of new concept of Limiting Phase Trajectories (LPT), without any restriction on the amplitudes of oscillation. As a result the conditions of intense energy exchange between the pendula and transition to energy localization are revealed in all possible diapason of initial conditions. By doing so, the roots and the domain of chaotic behavior are clarified as they are associated with this transition while simultaneously approaching the pendulum separatrix. The analytical findings are corroborated by numerical simulations. By considering the simplest case of two weakly coupled pendula, we pave the ground for new opening possibilities of significant extensions in both fundamental and applied directions. [7]. The majority of the results in all these fields relate to stationary dynamics and are based on the fundamental stationary regimes, namely the Nonlinear Normal Modes (NNMs) in finite systems [8,9] and solitons (breathers) in infinite models [1,2]. As for non-stationary processes, NNMs can also be used for their description provided that the intermodal resonance is absent [10]. However, the non-stationary resonance dynamics of finite systems turns out to be much more complicated. Therefore only isolated and predominantly numerical results were obtained in this field [11,12]. The recently developed concept of Limiting Phase Trajectories (LPTs) allowed for a systematic approach to description of non-stationary resonance regimes [13][14][15][16][17][18], including coupled pendula dynamics [19]. This concept introduces a fundamental non-stationary process of new type which corresponds to maximum possible energy exchange between the oscillators (in particular, pendula) or clusters of oscillators. In essence, the LPTs play in the non-stationary resonance dynamics of finite systems the role similar to that of the NNMs in the stationary theory and in the study of non-stationary yet non-resonant regimes. In terms of LPT, the transition from intense energy exchange between some clusters of oscillators (coherence domains), in particular, weakly coupled pendula, to energy localization in the initially excited cluster (oscillator) can also be predicted [11,20,21].However, all existing analytical results in the nonstationary resonance dynamics of finite-dimensional systems relate to some asymptotic limits which are either quasi-linear systems or separated oscillators (pendula) [11,14]. In this study we remove these restrictions and admit arbitrary oscillation amplitude in the framework of LPT concept. Assuming weak coupling, only the closeness to inter-...

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Non-smooth approximations of the limiting phase trajectories for the Duffing oscillator near 1:1 resonance

Cited by 55 publications

References 15 publications

Response enhancement in an oscillator chain

Response enhancement in an oscillator chain

Oscillatory chain with elastic supports and bending stiffness under conditions close to acoustic vacuum

Non-stationary resonance dynamics of weakly coupled pendula

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