Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / A / 4

Treat $t,T$ as independent variables, with $T={ϵ}^{2}t$. The chain rule givesTherefore the method of multiple scales isThe unspecified $O({ϵ}^4)$ term includes the higher correction in $a$. Only the displayed terms through order ${ϵ}^2$ are needed to determine the leading slow-amplitude system.

At order one, $({\partial }_t^2+1)y_0=0$, soAt order $ϵ$,The operator sends a constant to itself and sends $\cos 2t,\sin 2t$ to minus three times themselves. HenceAn additional homogeneous $\cos t,\sin t$ contribution can be absorbed into the definitions of $A,B$. There is no resonance forcing at this order; the first amplitude evolution therefore occurs on the chosen slower scale $T={ϵ}^{2}t$.

The order-${ϵ}^2$ equation isUsing $\cos t\cos 2t=(\cos 3t+\cos t)/2$ and $\cos t\sin 2t=(\sin 3t+\sin t)/2$, its resonance coefficients aresecular term are removed by setting these to zero:The eigenvalues of this two-by-two system satisfyThey are real with opposite signs for $-1/12<a_2<5/12$. Generic initial amplitudes then contain an exponentially growing slow-time component. At either leading endpoint the matrix is nonzero nilpotent matrix, so suitable initial amplitudes give linear slow-time growth; higher-order detuning decides the precise corrected boundary. Outside this interval the leading eigenvalues are purely imaginary and the slow amplitudes remain oscillatory.

Substitution into $a=1+a_2{ϵ}^2+\cdots$ gives the requested leading instability interval“Can become unbounded” permits special initial amplitudes without growth. The calculation identifies the second instability tongue of a weak Mathieu oscillator and its slow amplitudes for the second Mathieu instability tongue; it is not an assertion that every solution grows throughout the closed interval.