Use of degeneration to stabilize near grazing periodic motion in impact oscillators

In controlling the discontinuous grazing bifurcations in impact oscillators, a discrete-in-time linear feedback control strategy in the existing literature was used to change the conditions at the grazing point based on the grazing stability criterion. Though this strategy is effective for its linear inequality constraint of the control parameter domain, a smooth and predictable bifurcating response cannot be obtained for the controlled system, but the grazing induced chaos or period-adding phenomena. To improve this control strategy and stabilize the elementary near-grazing impact periodic motion in impact oscillators, one feasible control criterion is established in this paper by performing the perturbation analysis of the eigenvalues of the Jacobian matrix. It is found that the degeneration of both eigenvalues and grazing bifurcation can stabilize the elementary near-grazing impact periodic motion and eliminate the discontinuous jump phenomenon at grazing.