Stabilization of the saddle‐type periodic points in phase‐locked loop equation by the OGY‐control method

Abstract

AbstractIt is well known that the differential equation representing the phase‐locked loops (PLL) can produce chaotic attractor. In particular, since the periodic solution of the second type representing the out‐of‐lock condition can bifurcate to become chaos easily by small external force so its theoretical treatment is possible. Moreover, much research activity has been performed in this area and various interesting points have been confirmed by laboratory experiments.This paper is an attempt to stabilize various saddletype periodic orbits included in the chaotic attractor bifurcated from second‐type periodic solutions by the OGY method. In general, a chaotic attrrator includes a number of saddle‐type periodic orbits, and if one wants to stabilize one of them by the OGY method, one must know various values, e.g., coordinates and eigenvalues, etc., of the periodic points on the Poincaré map. However, there is no method to determine these values, in general.The authors obtained the saddle‐type 1‐, 2‐, 3‐, 4‐, and 8‐periodic orbits included in the chaotic attractor by using the continuation method. By applying the continuation method to the 1‐parameter bifurcation diagram, all parameter values required in the OGY method were determined and stabilization of any one of them was dethonstrated by computer simulation. In particular, the authors tested two OGY methods for N‐periodic orbits. One method needs one change of the control parameter after N mappings of external force, and the second method changes the parameter at every mapping. It was confirmed that the latter method takes a shorter time to achieve control.