Harmonic and subharmonic synchronization of a Rayleigh pendulum (Q909850)

The motion of the Rayleigh pendulum is described by the differential equation \[ (1)\quad \frac{d^ 2\theta}{dt^ 2}+\omega^ 2_ 0\theta =\lambda \omega_ 0\theta [a-\frac{4}{3}\frac{B}{\omega^ 2_ 0}(\frac{d\theta}{dt})^ 2]-2\lambda h\omega^ 2_ 0 \cos \Omega t \] where \((a,b,h,\lambda,\omega_ 0,\Omega)\in R^ 6\), \(ab^{-1}>0\), \(h\geq 0\), \(0<\lambda \leq 1\). In the paper the harmonic and subharmonic solutions of this equation are investigated. The method of averaging is employed. First and second order approximations in the cases \(\Omega - \omega_ 0\cong O(\lambda)\), \(\Omega -3\omega_ 0\cong O(\lambda)\), \(\Omega -5\omega_ 0\cong O(\lambda)\) are discussed.