Stabilizing and tracking unstable focus points in chaotic systems using a neural network

Abstract

We present a control system containing a neural network for stabilizing chaotic orbits on an unstable focus point embedded within a chaotic attractor. This control system does not require the location of the unstable focus point and the local linearized dynamics at the point. Even if a parameter of the chaotic system changes slowly, the control system can automatically track the unstable focus point both within and outside the chaotic regimes. We test the control system using a nonlinear map in numerical experiments.