Control of Neimark-Sacker bifurcation for a three-degree-of-freedom vibro-impact system with clearances

Abstract

This paper designs an impulsive state feedback control for a three-degree-of-freedom vibro-impact system with clearances to create a stable Neimark-Sacker bifurcation. The linear control gains of the feedback control are determined by using the explicit criteria of Neimark-Sacker bifurcation without directly using eigenvalues, which are designed to guarantee the existence of Neimark-Sacker bifurcation. The non-linear control gains are determined by using center manifold reduction theory and normal form approach, which are selected to govern the direction and stability of Neimark-Sacker bifurcation. The amplitude of the created invariant cycle from the Neimark-Sacker bifurcation is analytically obtained to achieve the control of the amplitude by selecting appropriate non-linear control gains. Numerical experiments are provided to show the effectiveness of the proposed control method.