Anti-controlling Neimark-Sacker bifurcation of a three-degree-of-freedom vibration system with clearance

Abstract

Anti-control of bifurcation, as an inverse problem of conventional bifurcation analysis, is aimed at creating a certain bifurcation with desired dynamic properties at a pre-specified system parameter location via control. The main purpose of this paper is to address the problem of anti-control of Neimark-Sacker bifurcation of a three-degree-of-freedom vibro-impact system with clearance (i.e., the second Hopf bifurcation of the original system), which may be viewed as a design approach to creating a quasi-periodic impact motion (or torus solution) at a specified system parameter location via control. Firstly, in the premise of no change of periodic solutions of the original system, when the difficulties that are brought about by the implicit Poincaré map of the vibro-impact system are considered, a linear feedback controller is added to the original system and a six-dimensional Poincaré map of the close-loop control system is established. In order to design a desired bifurcation solution by control, the multiple control gains are used to tune the existence of this bifurcation based on the corresponding critical criterion. However, for six-dimensional map of the vibro-impact system in the paper, the analytical expressions of all eigenvalues of Jacobi matrix with respect to parameters are unavailable. This implies that when the classical critical criterion described by the properties of eigenvalues is used, we have to numerically compute eigenvalues point by point and check their properties to search for the control gains. So, the numerical calculation is a laborious job in the process of determining the control gains. To overcome the difficulty originating from the classical bifurcation criterion, the explicit critical criterion without using eigenvalue calculation of high-dimensional map is used to obtain the controlling parameters area when quasi-periodic impact motion occurs. Then, the stability of quasi-periodic bifurcation solution is analyzed by utilizing the center manifold and normal formal theory. Finally the numerical experiments verify that the stable quasi-periodic impact motion can be generated at a designated system parameter point by the proposed control.