Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under OGY-based state-feedback control law: Order, chaos and exhibition of the border-collision bifurcation

This paper is concerned with the investigation of the nonlinear dynamic behavior of a one-degree-of-freedom (1-DoF) impact mechanical oscillator subject to a single rigid constraint and under an OGY-based state-feedback control law. Our analysis is mainly carried out through bifurcation diagrams. Several cases of the raised behaviors are also illustrated through time-traces, phase portraits and Poincaré sections. To prevent some problems in the computation of unstable solutions and their continuation to locate and characterize local bifurcations, we develop an analytical expression of a stroboscopic controlled hybrid Poincaré map. Moreover, we present conditions for determining the fixed point of such Poincaré map and for studying its stability. Through the hybrid Poincaré map, we analyze the displayed nonlinear phenomena in the controlled impacting oscillator. We show that several interesting behaviors are revealed including the period-doubling route to chaos, the period-adding cascade, interior and boundary crisis, the complete and incomplete chaotic chattering, the cyclic-fold bifurcation, the saddle-saddle bifurcation, the Neimark-Sacker bifurcation, the sub-critical period-doubling bifurcation, the grazing bifurcation, among others. Furthermore, we show also that the 1-DoF impacting mechanical oscillator displays, for the first time, the border-collision bifurcation, which is exhibited due to the OGY-based state-feedback control. In addition, we establish conditions for the localization of the border-collision bifurcation. Its occurrence is investigated via a two-parameter bifurcation diagram.