Sub- and superharmonic synchronization in weakly nonlinear systems: Integral constraints and duality

A forced system described by the differential equation: \[ x” + \epsilon f\left ( {x,x’} \right ) + \omega _0^2x = F\cos \omega t\] is considered for cases where $\left ( {n/m} \right )\omega$ is close to ${\omega _0}$ . (Here $m$ and $n$ are integers and $n/m > 1$ denotes superharmonic while $n/m < 1$ denotes subharmonics.) If the unforced system $\left ( {F = 0} \right )$ is conservative, the forced system is shown to possess an integral constraint and the solution is reduced to quadratures, even though the force adds or removes energy from the oscillations. Furthermore, the sub- and superharmonic cases where the $n/m$ ratios are inverse are shown to be intimately related, and results for one can be deduced from the other by appropriate interchange of variables.