Multiple-scale analysis: Difference between revisions

Revision as of 06:58, 9 June 2015

Revision as of 07:05, 9 June 2015

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===Coordinate transform to amplitude/phase variables===

We seek aA solution <math>y\approx r\cos\theta</math> is sought in new coordinates <math>(r,\theta)</math> where the amplitude <math>r(t)</math> varies slowly and the phase <math>\theta(t)</math> varies at an almost constant rate, namely <math>d\theta/dt\approx 1</math>.

Straightforward algebra finds the coordinate transform{{citation needed}}

:<math>y=r\cos\theta +\frac1{32}\varepsilon r^3\cos3\theta +\frac1{1024}\varepsilon^2r^5(-21\cos3\theta+\cos5\theta)+\mathcal O(\varepsilon^3)</math>

transforms Duffing's equation into the pair that the radius is constant <math>dr/dt=0</math> and the phase evolves according to

:<math>\frac{d\theta}{dt}=1 +\frac38\varepsilon r^2 -\frac{15}{256}\varepsilon^2r^4 +\mathcal O(\varepsilon^3).</math>

That is, Duffing's oscillations are constant amplitude but a different frequencies depending upon the amplitude.<ref>{{citation |first=A.J. |last=Roberts |title=Modelling emergent dynamics in complex systems |url=http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php |accessdate=2013-10-03 }}</ref>

More difficult examples are better treated using a time -dependent coordinate transform involving complex exponentials (as also invoked in the previous multiple time-scale approach). A web service will perform the analysis for a wide range of examples.<ref>{{citation |first=A.J. |last=Roberts |title=Construct centre manifolds of ordinary or delay differential equations (autonomous) |url=http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php |accessdate=2013-10-03 }}</ref>

==See also==