Targeting in chaos using analytically described clusters

Abstract

The OGY method provides a simple but powerfbl approach of controlling chaotic dynamics. This method can stabilise inherently unstable equilibrium modes of dissipative chaotic systems under the lack of knowledge about the system equations. However, it has the typical drawback of a long waiting time until the system starting from random initial conditions enters the close neighbourhood of the equilibrium mode to be stabilised, where the controller can be activated. The reduction of this drawback is known under the name of targeting. The Extended Control Regions method is a targeting approach, which can operate under the Iakof knowledge about the system equations by employing local models of the system dynamics extracted from empirical data. The method is based on the idea of identifiing and modelling those regions of the phase space, starting fi-om which the system can be steered to a close neighbourhood of the target within a few steps applying sinall perturbatiotns in the control parameters. So far, the modelling of the system dynamics within these phase space regions have been realised using artificial neural networks. In this study, two different strategies are developed in order to realise the clustered version of the Extended Control Regions method on basis of simple analytical models rather than neural networks. Each cluster obtained the gathered data is analytically described as a hyper-ellipsoid. Subsequently, the analytical models of the clusters are used for targeting purposes by applying small discrete variations in the control parameter. Simulation results on several chaotic systems with single control parameter show that the proposed method can achieve targeting using less memory and computation time than the Clustered Extended Control Regions method on cost of a slower targeting performance.