Neural models of arbitrary chaotic systems: construction and the role of time delayed feedback in control and synchronization

This paper proposes a simple methodology to construct an iterative neural network which mimics a given chaotic time series. The methodology uses the Gamma test to identify a suitable (possibly irregular) embedding of the chaotic time series from which a one step predictive model may be constructed. A one-step predictive model is then constructed as a feedforward neural network trained using the BFGS method. This network is then iterated to produce a close approximation to the original chaotic dynamics. We then show how the chaotic dynamics may be stabilized using time-delayed feedback. Delayed feedback is an attractive method of control because it has a very low computational overhead and is easy to integrate into hardware systems. It is also a plausible method for stabilization in biological neural systems. Using delayed feedback control, which is activated in the presence of a stimulus, such networks can behave as an associative memory, in which the act of recognition corresponds to stabilization onto an unstable periodic orbit. Surprisingly we find that the response of such systems is remarkably robust in the presence of noise. We briefly investigate the stability of the proposed control method and show that whilst the control/synchronisation methods are not always stable in the classical sense they may instead ∗Department of Computer Science, University of Wales, Cardiff, PO Box 916, Cardiff CF2 3XF, Wales, UK †Cognos Ltd. (London office), 6th Floor, King’s House, 10 Haymarket, London SW1Y 4BP, U.K. ‡Centre for Nonlinear Dynamics, & its Applications, University College London, London WC1E 6BT, England, UK be probabilistically locally stable. We also show how two independent copies of such a chaotic iterative network may be synchronized using variations of the delayed feedback method. Although less biologically plausible, these techniques may have interesting applications in secure communications.