Multiple-scale analysis: Difference between revisions

Revision as of 18:30, 21 April 2024

Revision as of 21:49, 7 May 2024

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As an example for the method of multiple-scale analysis, consider the undamped and unforced [[Duffing equation]]:<ref>This example is treated in: Bender & Orszag (1999) pp. 545–551.</ref>

<math display="block">\frac{d^2 y}{d t^2} + y + \varepsilon y^3 = 0,</math> <math display="block">y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>

which is a second-order [[ordinary differential equation]] describing a [[nonlinear system|nonlinear]]{{disambiguation needed|date=April 2024}} [[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 display="block">\frac{dp}{dt}=-\frac{\partial H}{\partial q}, \qquad \frac{dq}{dt}=+\frac{\partial H}{\partial p}, \quad \text{ with } \quad H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \varepsilon q^4,</math>

with ''q''&nbsp;=&nbsp;''y''(''t'') and ''p''&nbsp;=&nbsp;''dy''/''dt''. Consequently, the Hamiltonian ''H''(''p'',&nbsp;''q'') is a conserved quantity, a constant, equal to ''H''&nbsp;=&nbsp;{{sfrac|1|2}}&nbsp;+&nbsp;{{sfrac|1|4}}&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''y'' and ''dy''/''dt'' have to be bounded: