Perturbative Methods for Modelling (M2)

Section 11.4, p. 560

We use the multiscale analysis $\fs2 y(t)\sim Y_0(t, \epsilon t)$ for $\fs2 \epsilon\to 0$ with $\fs2 Y_0(t,\tau) =A(\tau) e^{i\,{t\over 2}}+ A^*(\tau) e^{-i\, \theta(\tau)} \, e^{-i\,{t\over2}}$ for the Mathieu
equation $\fs2 y''+\left[{1\over 4} + (a_1+2\, \cos t)\, \epsilon\right] y=0$. Show that $\fs2 |a_1|=1$ is the limit for the stability of the equilibrium $\fs2 y=0$.