Multiple-scale analysis: Difference between revisions

Revision as of 02:20, 10 August 2009

Revision as of 02:21, 10 August 2009

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:<math>\frac{d^2 y}{d t^2} + y + \epsilon y^3 = 0,</math> {{pad|3em}} <math>y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>

which is a second-order [[ordinary differential equation]] describing a [[nonlinear]] [[oscillator]]. A solution ''y''(''t'') is sought for small values of the (positive) nonlinearity parameter 0&nbsp;<&nbsp;''ε''&nbsp;≪&nbsp;1. The undamped Duffing equation is known to be a [[Hamiltonian system]]:

:<math>\frac{dp}{dt}=-\frac{dH}{dq}, \qquad \frac{dq}{dt}=+\frac{dH}{dp},</math> {{pad|2em}} with {{pad|2em}} <math>H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \epsilon q^4,</math>