Perturbative Methods for Modelling (M2)

Section 11.2, p. 549

Example 2, p. 554

We use the multiscale analysis $\fs2 y(t)\sim Y_0(t, \epsilon t)$ for $\fs2 \epsilon\to 0$ with $\fs2 Y_0(t,\tau) =R(\tau)\left[ e^{i\, \theta(\tau)} \, e^{i\,t} + e^{-i\, \theta(\tau)} \, e^{-i\,t}\right]$. Show the following relations:

(a) Duffing equation: $\fs2 y''+y+\epsilon\,y^3=0$:  $\fs2 \frac{dR}{d\tau}=0$ and $\fs2 \frac{d\theta}{d\tau}=3\,R^2/2$.

(b) Rayleigh oscillator: $\fs2 y''+ y=\epsilon\,\left[y'-{1\over3}\,(y')^3\right]$:  $\fs2 2\,\frac{dR}{d\tau}=R-R^3$ and $\fs2 \frac{d\theta}{d\tau}=0$.

Draw the shape of these solutions.