Evolving artificial neural networks to control chaotic systems Eric R. Weeks* and John M. Burgess ² Center for Nonlinear Dynamics and Department of Physics, University of Texas at Austin, Austin, Texas 78712 Received 7 April 1997 We develop a genetic algorithm that produces neural network feedback controllers for chaotic systems. The algorithm was tested on the logistic and He ´non maps, for which it stabilizes an unstable fixed point using small perturbations, even in the presence of significant noise. The network training method D. E. Moriarty and R. Miikkulainen, Mach. Learn. 22, 11 1996 requires no previous knowledge about the system to be controlled, including the dimensionality of the system and the location of unstable fixed points. This is the first dimension- independent algorithm that produces neural network controllers using time-series data. A software implemen- tation of this algorithm is available via the World Wide Web. S1063-651X9705308-7 PACS numbers: 05.45.+b, 07.05.Mh I. INTRODUCTION Chaotic behavior in a dynamical system can be sup- pressed by periodically applying small, carefully chosen per- turbations, often with the goal of stabilizing an unstable pe- riodic orbit of the system 1–3. Control of this type has been successfully applied to many experimental systems 3–7. In this paper we demonstrate a robust method of train- ing neural networks to control chaos. The method makes no assumptions about the system; the training algorithm oper- ates without knowledge of either the dimensionality of the dynamics or the location of any unstable fixed points. The structure of the controller is not fixed and the neural network is free to adopt nonlinear forms. After reviewing previous work in Sec. II, we present the details of our method in Sec. III. We use a modified version of Symbiotic Adaptive Neuro-Evolution SANE, developed by Moriarty and Miikkulainen 8–10. This method uses ge- netic algorithms to create neural networks with desirable characteristics, in this case the ability to stabilize unstable fixed points of a Poincare ´ map. SANE has proved to be fast and efficient in a variety of cases unrelated to chaos control 8–10. In Sec. IV we present results for control of the one- dimensional logistic map and the two-dimensional He ´ non map. In Sec. V we discuss extensions of our method and conclusions. II. PREVIOUS WORK A commonly applied method for control of chaotic dy- namical systems was discovered by Ott, Grebogi, and Yorke OGY1. The OGY method requires an analytical descrip- tion of the linear map describing the behavior near the fixed point 2. This map is used to determine small perturbations that, when periodically applied, use the system’s own dy- namics to send the system towards an unstable fixed point. Continued application of these perturbations keeps the sys- tem near the fixed point, thereby stabilizing the unstable fixed point even in a noisy system. The OGY method generally is inadequate when the sys- tem is far from the fixed point and the linear map is no longer valid. The original OGY method is also limited to controlling only one- or two-dimensional maps. However, the method has been extended to higher dimensions 11–13 and to cases where multiple control parameters are available 14. Several other analytical methods exist for controlling chaos. Hunt developed the method of occasional propor- tional feedback, a modification of the OGY algorithm 5. Pyragas developed a method that provides small control per- turbations for continuous systems 15. This elegant method uses a linear feedback based on an external signal or time- delayed feedback and requires little analytical preparation. Hu ¨ bler reported on the ‘‘dynamical key’’ method in which natural system dynamics may be time reversed and con- verted into perturbations to drive the system back to an un- stable fixed point 16. Petrov and Showalter presented a nonlinear method that relies on the construction of an extended-dimension, nonlinear control surface effectively a lookup table by taking into account the final state of the system after the application of a perturbation 17. An alternative to analytical control algorithms involves the use of neural networks to discover possibly novel con- trol techniques. Neural networks are well known for provid- ing solutions to some complex problems, even when an ana- lytical solution cannot be found 18. Neural networks have been used to control chaos in various systems, including the logistic map 19–21, the He ´ non map 22,23, other maps 23, and continuous systems 24–27. Several of these methods require the perturbation to be large 21,23–26 in contrast to methods such as the OGY algorithm. In some methods additional computation is required, such as prepro- cessing to find the algorithm to train the network 22,26,27, postprocessing to translate the output of a network into a desired perturbation 20,21, or an additional ‘‘switch’’ to activate the network when the system is close to the fixed point 20,23. In some cases, the location of the fixed point must be precisely specified 19,20,22,27. In other cases, tar- get dynamics such as a limit cycle must be specified 21,25. In one case all of the system variables must be available and controllable, although several different maps were control- lable by this algorithm 23. *Electronic address: weeks@chaos.ph.utexas.edu ² Electronic address: jburgess@chaos.ph.utexas.edu PHYSICAL REVIEW E AUGUST 1997 VOLUME 56, NUMBER 2 56 1063-651X/97/562/153110/$10.00 1531 © 1997 The American Physical Society