Multiple-scale analysis: Difference between revisions

Revision as of 01:08, 12 April 2023

Revision as of 20:33, 23 January 2024

Line 79:

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

Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the [[Normal form (dynamical systems)|method of normal forms]], <ref>{{citation| first1=C.-H. |last1=Lamarque |first2=C. |last2=Touze |first3=O. |last3=Thomas |title=An upper bound for validity limits of asymptotic analytical approaches based on normal form theory |journal=[[Nonlinear Dynamics (journal)|Nonlinear Dynamics]] |pages=1931–1949|year=2012 |volume=70 |issue=3 |doi=10.1007/s11071-012-0584-y |hdl=10985/7473 |s2cid=254862552 |url=https://hal.archives-ouvertes.fr/hal-00880968/file/LSIS-INSM_nonli_dyn_2012_thomas.pdf }}</ref> as described next.

A 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|date=June 2015}}

Line 100:

==References==

{{ref beginrefbegin}}

*{{citation

| last1=Kevorkian | first1=J.

Line 124:

| publisher = Wiley–VCH Verlag | isbn = 978-0-471-39917-9

}}

{{ref endrefend}}

==External links==