Controlling Laser Chaos

In 1990 Ott, Grebogi and Yorke described an attractive method (OGY) whereby small time-dependent perturbation applied to a chaotic system allowed to stabilize unstable periodic orbits[1]. This method is applicable to experimental situations in which a priori analytical knowledge of the system is not available[2,3]. Their method assumes the dynamics of the system can be represented as arising from a nonlinear map (e.g., a return map). The iterates are then given by Xn+1 = F(Xn,p), where p is some accessible parameter of the system. To control chaotic dynamics one only needs to learn the local dynamics around the desired unstable periodic orbit (e.g., a fixed point Xn=XF) on the nonlinear map : especially, the derivatives with respect to p of the orbit location. When the motion is near the periodic orbit(Xn#XF), small appropriate temporal perturbations of the control parameter p allow to hold the motion on its unstable periodic orbits.