RESONANT DYNAMICS OF AN AUTOPARAMETRIC SYSTEM - A STUDY USING HIGHER-ORDER AVERAGING

The autoparametric system consisting of a pendulum attached to a prima ry spring-mass is known to exhibit 1:2 internal resonance, and amplitu de-modulated chaos under harmonic forcing conditions. First-order aver aging studies and an analysis of the amplitude dynamics predicts that the response curves of the system exhibit saturation. The period-doubl ing route to chaos is observed following a Hopf bifurcation to limit c ycles. However, to answer questions about the range of the small param eter epsilon (a function of the forcing amplitude) for which the solut ions are valid, and about the persistence of some unstable dynamical b ehavior, like saturation, higher-order non-linear effects need to be t aken into account. Second-order averaging of the system is undertaken to address these issues. Loss of saturation is observed in the steady- stale amplitude responses. The breaking of symmetry in the various bif urcation sets becomes apparent as a consequence of epsilon appearing i n the averaged equations. For larger epsilon, second-order averaging p redicts additional Pitchfork and Hopf bifurcation points in the single -mode response. For the response between the two Hopf bifurcation poin ts from the coupled-mode solution branch, the period-doubling as well as the Silnikov mechanism for chaos are observed. The predictions of t he averaged equations are verified qualitatively for the original equa tions.