Parametric resonance

Parametric resonance occurs when periodic variation of a system parameter couples oscillatory modes and produces exponential amplitude growth. Multiple-scale solvability identifies the instability tongues and their detuning widths.

For $u^{\prime \prime }+[1+ϵ\sin t+{ϵ}^{2}f]u=0$, the second parametric-instability tongue is $-1/12<f<5/12$ to leading order. Outside its closure every leading slow amplitude is oscillatory.