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Resonant dynamics of an autoparametric system: A study using higher-order averaging

International Journal of Non-Linear Mechanics
Publication Date
DOI: 10.1016/0020-7462(95)00041-0


Abstract The autoparametric system consisting of a pendulum attached to a primary spring-mass is known to exhibit 1:2 internal resonance, and amplitude-modulated chaos under harmonic forcing conditions. First-order averaging studies and an analysis of the amplitude dynamics predicts that the response curves of the system exhibit saturation. The period-doubling route to chaos is observed following a Hopf bifurcation to limit cycles. However, to answer questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid, and about the persistence of some unstable dynamical behavior, like saturation, higher-order non-linear effects need to be taken into account. Second-order averaging of the system is undertaken to address these issues. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. For larger ε, second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response. For the response between the two Hopf bifurcation points from the coupled-mode solution branch, the period-doubling as well as the Silnikov mechanism for chaos are observed. The predictions of the averaged equations are verified qualitatively for the original equations.

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