Multiple-scale analysis: Difference between revisions

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In [[mathematics]] and [[physics]], '''multiple-scale analysis''' (also called the '''method of multiple scales''') comprises techniques used to construct uniformly valid [[approximation]]s to the solutions of [[perturbation theory|perturbation problems]], both for small as well as large values of the [[independent variable]]s. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) [[secular variation|secular terms]]. The latter puts constraints on the approximate solution, which are called '''solvability conditions'''.

Mathematics research from about the 1980s proposes{{Citation needed|date=December 2024}} that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see [[center manifold]] and [[slow manifold]]).

==Example: undamped Duffing equation==

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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]] [[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 ''H₀''&nbsp;=&nbsp;½{{sfrac|1|2}}&nbsp;+&nbsp;¼{{sfrac|1|4}}&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''yq'' and ''dy''/''dtp'' have to be bounded:

<math display="block">\left| y(t)q \right| \le \sqrt{1 + \tfrac12 \varepsilon} \quad \text{ and } \quad \left| \frac{dy}{dt}p \right| \le \sqrt{1 + \tfrac12 \varepsilon} \qquad \text{ for all } t.</math>The bound on q is found by equating H with p = 0 to H₀: <math>\tfrac12 q^2 + \tfrac14 \varepsilon q^4 = \tfrac12 + \tfrac14 \varepsilon</math>, and then dropping the q⁴ term. This is indeed an upper bound on |q|, though keeping the q⁴ term gives a smaller bound with a more complicated formula.

===Straightforward perturbation-series solution===

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<math display="block">\left[ -3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} \right]\, e^{+it} - A^3\, e^{+3it} + c.c.</math>

where ''c.c.'' denotes the [[complex conjugate]] of the preceding terms. The occurrence of ''secular terms'' can be prevented by imposing on the – yet unknown – amplitude ''A''(''t''₁) the ''solvability condition''

<math display="block">-3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} = 0.</math>,

i.e the amplitude equation:

<math display="block"> i \frac{dA}{dt_1} = - \frac{3}{2} |A|^2 A.</math>

The solution to the solvability condition, also satisfying the initial conditions {{math|1=''y''(0) = 1}} and {{math|1=''dy''/''dt''(0) = 0}}, is:

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===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 |bibcode=2012NonDy..70.1931L |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}}

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That is, Duffing's oscillations are of constant amplitude <math>r</math> but have different frequencies <math>d\theta/dt</math> 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.{{When|date=July 2024}}<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==

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| year = 2004

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

}}

* {{citation

| title = Physics of Wave Turbulence

| first = S. | last = Galtier

| year = 2023

| publisher = Cambridge University Press | isbn = 978-1-009-27588-0

}}

{{refend}}