Abstract
The logistic map has been used to describe period doubling bifurcations for periodically modulated lasers. It also represents an asymptotic approximation of Ikeda's map for a passive ring cavity. Because various control methods have been used recently to stabilize branches of periodic solutions in lasers, we investigate the logistic map with a standard Ott, Grebogi and Yorke (OGY) control. We explore the structure of this map plus perturbations and find considerable modifications to its bifurcation diagram. In addition to the original fixed points, we find a new fixed point and new period doubling bifurcations. We show that for certain values of small perturbations the new fixed point of the perturbed logistic map is stable, while its original fixed point becomes unstable. Our analysis suggests that new branches of solutions may exist in lasers as a result of the feedback control.
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