Applications of the Mathieu equation

Ruby, Lawrence

doi:10.1119/1.18290

Cited by 146 publications

(87 citation statements)

References 2 publications

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“…In the context of the present work, this discussion is pertinent in the case γ ∼ 1 where we have stability at the linear level. While there are non-linear couplings between the oscillating scale factor and higher energy excitations, these couplings result in excitations only when the higher energy excitations are sufficiently close to an integer multiple of the frequency of the oscillating scale factor [25]. A priori, the higher energy excitations in this system are not integer multiples of the oscillating scale factor.…”

Section: Jhep02(2014)029mentioning

confidence: 99%

A simple harmonic universe

Graham

Horn

Kachru

et al. 2014

J. High Energ. Phys.

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We explore simple but novel bouncing solutions of general relativity that avoid singularities. These solutions require curvature k = +1, and are supported by a negative cosmological term and matter with −1 < w < −1/3. In the case of moderate bounces (where the ratio of the maximal scale factor a + to the minimal scale factor a − is O(1)), the solutions are shown to be classically stable and cycle through an infinite set of bounces. For more extreme cases with large a + /a − , the solutions can still oscillate many times before classical instabilities take them out of the regime of validity of our approximations. In this regime, quantum particle production also leads eventually to a departure from the realm of validity of semiclassical general relativity, likely yielding a singular crunch. We briefly discuss possible applications of these models to realistic cosmology.

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Section: Jhep02(2014)029mentioning

confidence: 99%

A simple harmonic universe

Graham

Horn

Kachru

et al. 2014

J. High Energ. Phys.

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“…(40) ein, erkennt man, dass der Vorfaktor α ein rein reeller und positiver Parameter ist, der in der Form …”

Section: Ordnung Ein Einziger Pol Zweiter Ordnung Und Die Funktion F(unclassified

Spindephasierung im Kroghschen Kapillarmodell des Myokards

Ziener¹,

Wr²,

Schlemmer³

2013

Fortschr Röntgenstr

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“…The Mathieu equation is a special case of a linear second-order homogeneous differential equation with time-dependent coefficients, and appears in many applications in physics and engineering, specially in the description of parametrically driven systems, 22 as the systems considered in this work.…”

Section: Microbubble Dynamicsmentioning

confidence: 99%

Macroscopic acousto-mechanical analogy of a microbubble

Chaline

Jiménez

Mehrem

et al. 2015

The Journal of the Acoustical Society of America

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Microbubbles, either in the form of free gas bubbles surrounded by a fluid or encapsulated bubbles used currently as contrast agents for medical echography, exhibit complex dynamics under specific acoustic excitations. Nonetheless, considering their micron size and the complexity of their interaction phenomenon with ultrasound waves, expensive and complex experiments and/or simulations are required for their analysis. The behavior of a microbubble along its equator can be linked to a system of coupled oscillators. In this study, the oscillatory behavior of a microbubble has been investigated through an acousto-mechanical analogy based on a ring-shaped chain of coupled pendula. Observation of parametric vibration modes of the pendula ring excited at frequencies between 1 and 5 Hz is presented. Simulations have been carried out and show mode mixing phenomena. The relevance of the analogy between a microbubble and the macroscopic acousto-mechanical setup is discussed and suggested as an alternative way to investigate the complexity of microbubble dynamics.

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Applications of the Mathieu equation

Abstract: of the start pulse in time of arrival correlators," Rev. Sci. lnstrum. 60, 634-637 (1989). 2 £Tor reasons of electronic simplicity, the delay is made to correspond to a number of pulses that is a power of 2.

Cited by 146 publications

References 2 publications

A simple harmonic universe

A simple harmonic universe

Spindephasierung im Kroghschen Kapillarmodell des Myokards

Macroscopic acousto-mechanical analogy of a microbubble

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