Controlling low-dimensional chaos: determination and stabilization of unstable periodic orbits by Kohonen neural nets
Michael Funke, Michael Herrmann, Ralf Der · International Journal of Adaptive Control and Signal Processing · 1997
A simple but efficient neural-network-based algorithm for non-linear control of chaotic systems is presented. The scheme relies on the method proposed by Ott et al. (Phys. Rev. Lett., 64, 1196 (1990)) to stabilize unstable periodic orbits by appropriate small changes in a control parameter. In contrast with this, our approach does not make use of an analytical description of the system evolution. The dynamics is evaluated by a self-organizing Kohonen network with an altered learning rule, which is able to learn the map of the system and to determine the positions of unstable periodic orbits of a given period. At the end of learning, a set of control neurons is generated which target the system along a quasi-optimal path towards the orbit. Besides its intrinsic tolerance against weak noise, the main advantage of the algorithm is its ability to take into account system constraints that occur in practical applications. The mean value of the control parameter and the range of allowed changes can be chosen in advance, and if more than one fixed point exists, the algorithm adapts to the most appropriate one concerning the control effort. © 1997 by John Wiley & Sons, Ltd.