Multiple-scale analysis: Difference between revisions

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===Differential equation and energy conservation===

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> {{pad|3em}} <math display="block">y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>

:<math>\frac{d^2 y}{d t^2} + y + \varepsilon y^3 = 0,</math> {{pad|3em}} <math>y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>

which is a second-order [[ordinary differential equation]] describing a [[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>

:<math>\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;½&nbsp;+&nbsp;¼&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''y'' and ''dy''/''dt'' have to be bounded:

:<math display="block">\left| y(t) \right| \le \sqrt{1 + \tfrac12 \varepsilon} \quad \text{ and } \quad \left| \frac{dy}{dt} \right| \le \sqrt{1 + \tfrac12 \varepsilon} \qquad \text{ for all } t.</math>{{pad|3em}}

:<math>\left| y(t) \right| \le \sqrt{1 + \tfrac12 \varepsilon} \quad \text{ and } \quad \left| \frac{dy}{dt} \right| \le \sqrt{1 + \tfrac12 \varepsilon} \qquad \text{ for all } t.</math>{{pad|3em}}

===Straightforward perturbation-series solution===

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===Method of multiple scales===

To construct a solution that is valid beyond <math>t = O(\epsilon^{-1})</math>, the method of ''multiple-scale analysis'' is used. Introduce the slow scale ''t''₁:

:<math display="block">t_1 = \varepsilon t\,</math>

:<math>t_1 = \varepsilon t\,</math>

and assume the solution ''y''(''t'') is a perturbation-series solution dependent both on ''t'' and ''t''₁, treated as:

:<math display="block">y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.</math>

:<math>y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.</math>

So:

<math display="block"> \begin{align}

\frac{dy}{dt}

:<math>

&= \left( \frac{\partial Y_0}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_0}{\partial t_1} \right)

\begin{align}

+ \varepsilon \left( \frac{\partial Y_1}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_1}{\partial t_1} \right)

\frac{dy}{dt}

+ \cdots

&= \left( \frac{\partial Y_0}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_0}{\partial t_1} \right)

\\

+ \varepsilon \left( \frac{\partial Y_1}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_1}{\partial t_1} \right)

&= \frac{\partial^2 Y_0}{\partial t^2}

+ \cdots

+ \varepsilon \left( \frac{\partial Y_0}{\partial t_1} + \frac{\partial Y_1}{\partial t} \right)

\\

&=+ \fracmathcal{\partial Y_0O}{(\partial t} varepsilon^2),

\end{align}</math>

+ \varepsilon \left( \frac{\partial Y_0}{\partial t_1} + \frac{\partial Y_1}{\partial t} \right)

+ \mathcal{O}(\varepsilon^2),

\end{align}

</math>

using ''dt''₁/''dt''&nbsp;=&nbsp;''ε''. Similarly:

<math display="block">\frac{d^2 y}{d t^2}

= \frac{\partial^2 Y_0}{\partial t^2} + Y_0 &= 0,

:<math>

+ \varepsilon \left( 2 \frac{\partial^2 Y_0}{\partial t\, \partial t_1} + \frac{\partial^2 Y_1}{\partial t^2} \right)

\frac{d^2 y}{d t^2}

+ \mathcal{O}(\varepsilon^2),.</math>

= \frac{\partial^2 Y_0}{\partial t^2}

+ \varepsilon \left( 2 \frac{\partial^2 Y_0}{\partial t\, \partial t_1} + \frac{\partial^2 Y_1}{\partial t^2} \right)

+ \mathcal{O}(\varepsilon^2).

</math>

Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:

<math display="block">\begin{align}

\frac{\partial^2 Y_0}{\partial t^2} + Y_0 &= 0, \\

:<math>

\frac{\partial^2 Y_1}{\partial t^2} + Y_1 &= - Y_0^3 - 2\, \frac{\partial^2 Y_0}{\partial t\, \partial t_1}.

\begin{align}

\end{align}</math>

\frac{\partial^2 Y_0}{\partial t^2} + Y_0 &= 0,

\\

\frac{\partial^2 Y_1}{\partial t^2} + Y_1 &= - Y_0^3 - 2\, \frac{\partial^2 Y_0}{\partial t\, \partial t_1}.

\end{align}

</math>

===Solution===

The zeroth-order problem has the general solution:

:<math display="block">Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},</math>

:<math>Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},</math>

with ''A''(''t''₁) a [[complex-valued amplitude]] to the zeroth-order solution ''Y''₀(''t'',&nbsp;''t''₁) and ''i''²&nbsp;=&nbsp;−1. Now, in the first-order problem the forcing in the [[right hand side]] of the differential equation is

:<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>

:<math>\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>

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

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

:<math display="block">A = \tfrac12tfrac 1 2\, \exp \left(\tfrac38tfrac 3 8\, i \, t_1 \right).</math>

The solution to the solvability condition, also satisfying the initial conditions ''y''(0)&nbsp;=&nbsp;1 and ''dy''/''dt''(0)&nbsp;=&nbsp;0, is:

:<math>A = \tfrac12\, \exp \left(\tfrac38\, i \, t_1 \right).</math>

As a result, the approximate solution by the multiple-scales analysis is

:<math display="block">y(t) = \cos \left[ \left( 1 + \tfrac38\, \varepsilon \right) t \right] + \mathcal{O}(\varepsilon),</math>

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

This new solution is valid until <math>t = O(\epsilon^{-2})</math>. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, ''i.e.'':, {{math|1=''t''₂&nbsp; =&nbsp; ''ε''²&nbsp; ''t''}}, {{math|1=''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>

:<math>y(t) = \cos \left[ \left( 1 + \tfrac38\, \varepsilon \right) t \right] + \mathcal{O}(\varepsilon),</math>

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

This new solution is valid until <math>t = O(\epsilon^{-2})</math>. 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>

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

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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}}

:<math display="block">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>

:<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 display="block">\frac{d\theta}{dt} = 1 + \frac38frac 3 8 \varepsilon r^2 -\frac{15}{256}\varepsilon^2r^4 +\mathcal O(\varepsilon^3).</math>

:<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 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>