Differential equation and energy conservation

As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation:^[1]

$${\displaystyle {\frac {d^{2}y}{dt^{2}}}+y+\varepsilon y^{3}=0,}$$

$${\displaystyle y(0)=1,\qquad {\frac {dy}{dt}}(0)=0,}$$

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:

$${\displaystyle {\frac {dp}{dt}}=-{\frac {\partial H}{\partial q}},\qquad {\frac {dq}{dt}}=+{\frac {\partial H}{\partial p}},\quad {\text{ with }}\quad H={\tfrac {1}{2}}p^{2}+{\tfrac {1}{2}}q^{2}+{\tfrac {1}{4}}\varepsilon q^{4},}$$

with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H = ½ + ¼ ε for the given initial conditions. This implies that both y and dy/dt have to be bounded:

$${\displaystyle \left|y(t)\right|\leq {\sqrt {1+{\tfrac {1}{2}}\varepsilon }}\quad {\text{ and }}\quad \left|{\frac {dy}{dt}}\right|\leq {\sqrt {1+{\tfrac {1}{2}}\varepsilon }}\qquad {\text{ for all }}t.}$$

Straightforward perturbation-series solution

A regular perturbation-series approach to the problem proceeds by writing ${\textstyle y(t)=y_{0}(t)+\varepsilon y_{1}(t)+{\mathcal {O}}(\varepsilon ^{2})}$ and substituting this into the undamped Duffing equation. Matching powers of ${\textstyle \varepsilon }$ gives the system of equations

$${\displaystyle {\begin{aligned}{\frac {d^{2}y_{0}}{dt^{2}}}+y_{0}&=0,\\{\frac {d^{2}y_{1}}{dt^{2}}}+y_{1}&=-y_{0}^{3}.\end{aligned}}}$$

Solving these subject to the initial conditions yields

$${\displaystyle y(t)=\cos(t)+\varepsilon \left[{\tfrac {1}{32}}\cos(3t)-{\tfrac {1}{32}}\cos(t)-\underbrace {{\tfrac {3}{8}}\,t\,\sin(t)} _{\text{secular}}\right]+{\mathcal {O}}(\varepsilon ^{2}).}$$

Note that the last term between the square braces is secular: it grows without bound for large |t|. In particular, for ${\displaystyle t=O(\varepsilon ^{-1})}$ 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.

Method of multiple scales

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

$${\displaystyle t_{1}=\varepsilon t}$$

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

$${\displaystyle y(t)=Y_{0}(t,t_{1})+\varepsilon Y_{1}(t,t_{1})+\cdots .}$$

So:

$${\displaystyle {\begin{aligned}{\frac {dy}{dt}}&=\left({\frac {\partial Y_{0}}{\partial t}}+{\frac {dt_{1}}{dt}}{\frac {\partial Y_{0}}{\partial t_{1}}}\right)+\varepsilon \left({\frac {\partial Y_{1}}{\partial t}}+{\frac {dt_{1}}{dt}}{\frac {\partial Y_{1}}{\partial t_{1}}}\right)+\cdots \\&={\frac {\partial Y_{0}}{\partial t}}+\varepsilon \left({\frac {\partial Y_{0}}{\partial t_{1}}}+{\frac {\partial Y_{1}}{\partial t}}\right)+{\mathcal {O}}(\varepsilon ^{2}),\end{aligned}}}$$

using dt₁/dt = ε. Similarly:

$${\displaystyle {\frac {d^{2}y}{dt^{2}}}={\frac {\partial ^{2}Y_{0}}{\partial t^{2}}}+\varepsilon \left(2{\frac {\partial ^{2}Y_{0}}{\partial t\,\partial t_{1}}}+{\frac {\partial ^{2}Y_{1}}{\partial t^{2}}}\right)+{\mathcal {O}}(\varepsilon ^{2}).}$$

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

$${\displaystyle {\begin{aligned}{\frac {\partial ^{2}Y_{0}}{\partial t^{2}}}+Y_{0}&=0,\\{\frac {\partial ^{2}Y_{1}}{\partial t^{2}}}+Y_{1}&=-Y_{0}^{3}-2\,{\frac {\partial ^{2}Y_{0}}{\partial t\,\partial t_{1}}}.\end{aligned}}}$$

Solution

The zeroth-order problem has the general solution:

$${\displaystyle Y_{0}(t,t_{1})=A(t_{1})\,e^{+it}+A^{\ast }(t_{1})\,e^{-it},}$$

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

$${\displaystyle \left[-3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}\right]\,e^{+it}-A^{3}\,e^{+3it}+c.c.}$$

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

$${\displaystyle -3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}=0.}$$

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

$${\displaystyle A={\tfrac {1}{2}}\,\exp \left({\tfrac {3}{8}}\,i\,t_{1}\right).}$$

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

$${\displaystyle y(t)=\cos \left[\left(1+{\tfrac {3}{8}}\,\varepsilon \right)t\right]+{\mathcal {O}}(\varepsilon ),}$$

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

This new solution is valid until ${\displaystyle t=O(\epsilon ^{-2})}$. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., t₂ = ε² t, t₃ = ε³ t, etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999).^[2]

Coordinate transform to amplitude/phase variables

Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, ^[3] as described next.

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

$${\displaystyle y=r\cos \theta +{\frac {1}{32}}\varepsilon r^{3}\cos 3\theta +{\frac {1}{1024}}\varepsilon ^{2}r^{5}(-21\cos 3\theta +\cos 5\theta )+{\mathcal {O}}(\varepsilon ^{3})}$$

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

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

That is, Duffing's oscillations are of constant amplitude ${\displaystyle r}$ but have different frequencies ${\displaystyle d\theta /dt}$ depending upon the amplitude.^[4]

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.^[5]