Controlling Chaos with Artificial Neural Network: Numerical Studies and Experiments

Chaotic systems are highly susceptible to control by using small perturbations to a system constraint. Feedback methods have been applied to taming chaos in magnetoelastic and hydrodynamic systems, electric circuits, lasers, chemical reactions, and tissues of heart and brain in vitro.1 The well-known Ott-Grebogi-Yorke (OGY) algorithm2 and its variations are based on the notion that for controlling chaos one needs to know only the local dynamics in a small (linear) range around a fixed point (corresponding to the targeted unstable periodic orbit, UPO), on some n-dimensional surface-of-section (Poincaré section) of the phase space trajectories. The OGY method has been further developed by Petrov et al.3 and Rhode et al.4 They studied the dynamics of the system around the fixed point under the effect of random perturbations, then the goal dynamics has been targeted by using a control rule with empirically determined constants in a control formula. In experimental settings, however, application of these methods is often troublesome because of noise and shift in the system constraints.5 It seemed inevitable to develop a better technique so as to make chaos control a routine (automated) procedure. Artificial neural networks are widely used in chemometrics, especially when the evaluation of experimental data requires complex, nonlinear fitting. A comprehensive review on the topic has been published by Sumpter et al.6 It has been shown first by Alsing et al.7 that chaos control can be implemented by using a nonlinear fitting procedure with an ANN. In their numerical work, however, they just simply fitted the data to the OGY formula by an ANN and applied the trained network to control chaos. A strategy developed later by Lebender et al.8 takes advantage of the nonlinearity built in an ANN, and it also works outside the linear region of the fixed point. However, training of the ANN required an additional numerical fitting procedure. Konishi et al.9 have also developed an on-line chaos controller that, however, works only for fixed point with eigenvalues in a given range or it requires a complicated procedure to construct an error function for the ANN.10 The method of Bakker et al.11 fits a global model to discrete time series data and controls chaos using the trained network. This method has been experimentally applied to control the chaotic motion of a pendulum. These methods have not been tested on real chemical systems so far. Our goal was to develop ANN algorithms as simple as possible and test their effectiveness in experiments. This paper is structured as follows. First, a brief introduction to artificial neural networks is presented. Then we devise a simple strategy for controlling chaos by taking advantage of a well-known feature of artificial neural networks: they can “learn” the linear or nonlinear rules embedded even in a noisy data set. The proposed method is first tested for taming chemical chaos in a simple three-variable model, the chaotic Autocatalator.12 We show that after learning the map-based representation of the chaotic dynamics at a given range of an accessible control parameter, the trained network can be readily applied to controlling chaos based on the simple proportional feedback (SPF) algorithm by Peng et al.13 The network is then modified so as to implement the so-called recursive proportional feedback (RPF) method for controlling chaos.14 The improved ANN algorithm is applied to a model for the respiratory behavior of a diffusively coupled two-cell system.15,16 Finally, we test the suggested method on an experimental system, the chaotic electrodissolution of copper in concentrated phosphoric acid electrolyte.17-21