Multiple-scale analysis

File:Duffing-equation-perturbative-approaches.svg

As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation:This example is treated in: Bender & Orszag (1999) pp. 545–551.

$$\backslash frac\{d^2\; y\}\{d\; t^2\}\; +\; y\; +\; \backslash varepsilon\; y^3\; =\; 0,$$ $$y(0)=1,\; \backslash qquad\; \backslash frac\{dy\}\{dt\}(0)=0,$$

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 < ε ≪ 1. The undamped Duffing equation is known to be a Hamiltonian system:

$$\backslash frac\{dp\}\{dt\}=-\backslash frac\{\backslash partial\; H\}\{\backslash partial\; q\},\; \backslash qquad\; \backslash frac\{dq\}\{dt\}=+\backslash frac\{\backslash partial\; H\}\{\backslash partial\; p\},\; \backslash quad\; \backslash text\{\; with\; \}\; \backslash quad\; H\; =\; \backslash tfrac12\; p^2\; +\; \backslash tfrac12\; q^2\; +\; \backslash tfrac14\; \backslash varepsilon\; q^4,$$

with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H₀ = {{sfrac|1|2}} + {{sfrac|1|4}} ε for the given initial conditions. This implies that both q and p have to be bounded:

$$\backslash left|\; q\; \backslash right|\; \backslash le\; \backslash sqrt\{1\; +\; \backslash tfrac12\; \backslash varepsilon\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash left|\; p\; \backslash right|\; \backslash le\; \backslash sqrt\{1\; +\; \backslash tfrac12\; \backslash varepsilon\}\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; t.$$The bound on q is found by equating H with p = 0 to H₀: $\backslash tfrac12\; q^2\; +\; \backslash tfrac14\; \backslash varepsilon\; q^4\; =\; \backslash tfrac12\; +\; \backslash tfrac14\; \backslash varepsilon$, 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.

A regular perturbation-series approach to the problem proceeds by writing $y(t)\; =\; y\_0(t)\; +\; \backslash varepsilon\; y\_1(t)\; +\; \backslash mathcal\{O\}(\backslash varepsilon^2)$ and substituting this into the undamped Duffing equation. Matching powers of $\backslash varepsilon$ gives the system of equations

$$\backslash begin\{align\}$$

\frac{d^2 y_0}{dt^2} + y_0 &= 0,\\

\frac{d^2 y_1}{dt^2} + y_1 &= - y_0^3.

\end{align}

Solving these subject to the initial conditions yields

y(t) = \cos(t)

+ \varepsilon \left[ \tfrac{1}{32} \cos(3t) - \tfrac{1}{32} \cos(t) - \underbrace{\tfrac 3 8\, t\, \sin(t)}_\text{secular} \right]

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

Note that the last term between the square braces is secular: it grows without bound for large |t|. In particular, for $t\; =\; O(\backslash varepsilon^\{-1\})$ this term is O(1) and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution.

To construct a solution that is valid beyond $t\; =\; O(\backslash epsilon^\{-1\})$, the method of multiple-scale analysis is used. Introduce the slow scale t₁:

$$t\_1\; =\; \backslash varepsilon\; t$$

and assume the solution y(t) is a perturbation-series solution dependent both on t and t₁, treated as:

$$y(t)\; =\; Y\_0(t,t\_1)\; +\; \backslash varepsilon\; Y\_1(t,t\_1)\; +\; \backslash cdots.$$

So:

$$\backslash begin\{align\}$$

\frac{dy}{dt}

&= \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)

+ \cdots

\\

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

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

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

\end{align}

using dt₁/dt = ε. Similarly:

$$\backslash frac\{d^2\; y\}\{d\; t^2\}$$

= \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).

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

$$\backslash begin\{align\}$$

\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}

The zeroth-order problem has the general solution:

$$Y\_0(t,t\_1)\; =\; A(t\_1)\backslash ,\; e^\{+it\}\; +\; A^\backslash ast(t\_1)\backslash ,\; e^\{-it\},$$

with A(t₁) a complex-valued amplitude to the zeroth-order solution Y₀(t, t₁) and i² = −1. Now, in the first-order problem the forcing in the right hand side of the differential equation is

$$\backslash left[\; -3\backslash ,\; A^2\backslash ,\; A^\backslash ast\; -\; 2\backslash ,\; i\backslash ,\; \backslash frac\{dA\}\{dt\_1\}\; \backslash right]\backslash ,\; e^\{+it\}\; -\; A^3\backslash ,\; e^\{+3it\}\; +\; c.c.$$

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(t₁) the solvability condition

$$-3\backslash ,\; A^2\backslash ,\; A^\backslash ast\; -\; 2\backslash ,\; i\backslash ,\; \backslash frac\{dA\}\{dt\_1\}\; =\; 0.$$

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

$$A\; =\; \backslash tfrac\; 1\; 2\backslash ,\; \backslash exp\; \backslash left(\backslash tfrac\; 3\; 8\backslash ,\; i\; \backslash ,\; t\_1\; \backslash right).$$

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

$$y(t)\; =\; \backslash cos\; \backslash left[\; \backslash left(\; 1\; +\; \backslash tfrac38\backslash ,\; \backslash varepsilon\; \backslash right)\; t\; \backslash right]\; +\; \backslash mathcal\{O\}(\backslash varepsilon),$$

using {{math|1=t₁ = εt}} and valid for {{math|1=εt = O(1)}}. This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method.

This new solution is valid until $t\; =\; O(\backslash epsilon^\{-2\})$. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., {{math|1=t₂ = ε² t}}, {{math|1=t₃ = ε³ 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}}).Bender & Orszag (1999) p. 551.

Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms,{{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 |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 }} as described next.

A solution $y\backslash approx\; r\backslash cos\backslash theta$ is sought in new coordinates $(r,\backslash theta)$ where the amplitude $r(t)$ varies slowly and the phase $\backslash theta(t)$ varies at an almost constant rate, namely $d\backslash theta/dt\backslash approx\; 1.$ Straightforward algebra finds the coordinate transform{{citation needed|date=June 2015}}

$$y=r\backslash cos\backslash theta\; +\backslash frac1\{32\}\backslash varepsilon\; r^3\backslash cos3\backslash theta\; +\backslash frac1\{1024\}\backslash varepsilon^2r^5(-21\backslash cos3\backslash theta+\backslash cos5\backslash theta)+\backslash mathcal\; O(\backslash varepsilon^3)$$

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

$$\backslash frac\{d\backslash theta\}\{dt\}\; =\; 1\; +\; \backslash frac\; 3\; 8\; \backslash varepsilon\; r^2\; -\backslash frac\{15\}\{256\}\backslash varepsilon^2r^4\; +\backslash mathcal\; O(\backslash varepsilon^3).$$

That is, Duffing's oscillations are of constant amplitude $r$ but have different frequencies $d\backslash theta/dt$ depending upon the amplitude.{{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 }}

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}}{{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 }}

• Method of matched asymptotic expansions
• WKB approximation
• Method of averaging
• Krylov–Bogoliubov averaging method

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• {{scholarpedia | title=Multiple scale analysis | urlname=Multiple_scale_analysis | curator=Carson C. Chow}}

Category:Mathematical physics

Category:Asymptotic analysis

Category:Perturbation theory