Controlling Chaos in a Simple Nonlinear Fibre Resonator

In 1979, Ikeda [1] showed that an optical resonator containing a nonlinear optical element could not only have a bistable behaviour, but the system could also go unstable at certain power levels. Nonlinear optical fibre resonators can exhibit these Ikeda instabilities as well as the bistable behaviour [2-4]. Obviously, an unstable output is unwanted in a bistable device and methods need to be investigated to limit or remove the chaotic behaviour. One possible solution is to use a method for stabilizing unstable periodic points on the chaotic attractor by applying small perturbations of a control parameter, first proposed by Ott, Grebogi and Yorke (OGY) in 1990 [5], who stabilized an unstable point in the Hénon map, even in the presence of noise. We report here on the application of the OGY method to the Ikeda map, which can be shown to describe the iterative passage of the electric field around a simple nonlinear fibre resonator, see Fig. 1, [4]. The focus is on the fundamental possibility of controlling the Ikeda instabilities and not on the practicalities, which may be challenging.