Instabilities in a One-Dimensional Driven Bistable System under Delayed Feedback Control

We consider the dynamics of a delayed feedback controlled system, which is designed to stabilize an unstable homogeneous symmetric periodic state in a periodically driven bistable system in one dimension. The existence of a finite limit on our ability to measure spatial structure in the feedback system is taken into account by employing Fourier space filtering with a cutoff, which corresponds to the measurement resolution, in the wave number domain. The main purpose of this paper is to clarify the influences of the resolution on the limitation of control and the dynamics in various unstable regimes. First, we give an overview of the dynamics. The characteristic features of the dynamics due to the first and second instabilities, the dynamics in resonant regimes, and the dynamics in a chaotic regime are elucidated. Second, on the basis of a linear stability analysis, we estimate the stability criterion in a conventional control model and an extended control model. Third, we treat the transition from a standing wave state to a vibrating standing wave state in which the mean level of the standing wave undergoes a Hopf bifurcation, and obtain the borders of the transitions for the two controlled models.