Asymptotic analysis of the parametrically driven damped nonlinear evolution equation

Master Thesis

1997

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University of Cape Town

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Abstract
Singular perturbation methods are used to obtain amplitude equations for the parametrically driven damped linear and nonlinear oscillator, the linear and nonlinear Klein-Gordon equations in the small-amplitude limit in various frequency regimes. In the case of the parametrically driven linear oscillator, we apply the Lindstedt-Poincare method and the multiple-scales technique to obtain the amplitude equation for the driving frequencies Wdr ~ 2ω₀,ω₀, (2/3)ω₀ and (1/2)ω₀. The Lindstedt-Poincare method is modified to cater for solutions with slowly varying amplitudes; its predictions coincide with those obtained by the multiple-scales technique. The scaling exponent for the damping coefficient and the correct time scale for the parametric resonance are obtained. We further employ the multiple-scales technique to derive the amplitude equation for the parametrically driven pendulum for the driving frequencies Wdr ~ 2ω₀, ω₀, (2/3)ω₀, (1/2)ω₀ and 4ω₀. We obtain the correct scaling exponent for the amplitude of the solution in each of these frequency regimes.
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Bibliography: pages 179-184.

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