Multiple-scale analysis: Difference between revisions

Revision as of 01:24, 3 October 2013

Revision as of 22:17, 3 October 2013

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using ''t''₁ = ''εt'' and valid for ''εt'' = O(1). This agrees with the nonlinear [[frequency]] changes found by employing the [[Lindstedt–Poincaré method]].

Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, ''i.e.'': ''t''₂&nbsp;=&nbsp;''ε''²&nbsp;''t'', ''t''₃&nbsp;=&nbsp;''ε''³&nbsp;''t'', etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see {{harvnb|Kevorkian|Cole|1996}}; {{harvnb|Bender|Orszag|1999}}).<ref>Bender & Orszag (1999) p. 551.</ref>

Alternatively, modern sound approaches derive these sorts of models using coordinate transforms <ref>{{citation| first1=C.-H. |last1=Lamarque, |first2=C. |last2=Touze, and |first3=O. |last3=Thomas. |title=An upper bound for validity limits of asymptotic analytical approaches based on normal form theory. <em>|journal=Nonlinear Dynamics</em>, |pages=1931–1919 |year=2012 1|volume=70 |issue=3 |doi=10.1007/s11071-012-19,0584-y 2012.}}</ref> as also described next.

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

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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>

<ref>A. J. Roberts, <em>Modelling emergent dynamics in complex systems</em> [http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php]</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>A. J. Roberts, <em>Construct centre manifolds of ordinary or delay differential equations (autonomous)</em> [http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php]</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, <em>|title=Construct centre manifolds of ordinary or delay differential equations (autonomous)</em> [|url=http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php] |accessdate=2013-10-03 }}</ref>

==See also==