Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection

Sahai, Tuhin

doi:10.1007/s11071-010-9716-4

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…Systems with multiple time scales are prevalent in a wide variety of applications related to smart grids [38,39,40], building systems [28], and micromechanical oscillators [41,42,43], to name a few. Simulating these systems is challenging due to their inherent stiffness [44].…”

Section: Polynomial Chaos Based Uncertainty Quantification In Systems...mentioning

confidence: 99%

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Pasini¹,

Sahai²

2013

Preprint

Self Cite

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Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, has multiple time scales, or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of the order of the expansion. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the time scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

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Section: Polynomial Chaos Based Uncertainty Quantification In Systems...mentioning

confidence: 99%

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Pasini¹,

Sahai²

2013

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Polynomial chaos based uncertainty quantification in systems with multiple time scales. Systems with multiple time scales are prevalent in a wide variety of applications related to smart grids [13,29,39], building systems [38], and micromechanical oscillators [32,33,36], to name a few. Simulating these systems is challenging due to their inherent stiffness [7].…”

mentioning

confidence: 99%

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Pasini¹,

Sahai²

2014

Journal of Computational Dynamics

Self Cite

View full text Add to dashboard Cite

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

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Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities

Cho

Jeong

et al. 2012

International Journal of Solids and Structures

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Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection

Cited by 3 publications

References 27 publications

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities

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