Abstract
Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric system are studied for resonant excitations. The method of averaging is used to obtain first order approximations to the response of the system. In the subharmonic case of internal and external resonance, where the external excitation is in the neighborhood of the higher natural frequency, a complete bifurcation analysis of the averaged equations is undertaken. The “locked pendulum” mode of response bifurcates to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, though it requires mistuning from the exact internal resonance condition. The Hopf bifurcation sets are constructed and dynamic steady solutions of the amplitude or averaged equations are investigated using software packages AUTO and KAOS. It is shown that both super- and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.
