Control of periodic dynamics of nonlinear and chaotic discrete dynamical systems

Abstract

The present work consists of a generalization to dimensions greater than 1 of the results obtained on control and more precisely on the suppression of periodic rhythms in unidimensional nonlinear discrete dynamic systems. This method is based on predictive control which is known for the control of chaos. The suppression of the stable periodic orbit by predictive control requires a rigorous estimation of the control distance. Therefore, predictive control must be corroborated by an estimate of the size of the restricted attraction basin of the unstable fixed point. This clearly improves the work of T. Ushio and S. Yamamoto in which the control distance is arbitrary. The advantage of this mathematical approach is that it can be rigorously applied and the results validated not only to discrete multi-dimensional nonlinear chaotic systems (logistics map, Henon map) but also in the case of the prevention of some cardiac pathologies using the action potential duration map (APD), a nonlinear chaotic discrete dynamical system. The properties of restitution of the duration of the action potential and the alternation of repolarization are important arrhythmogenic factors. Thus, although the alternating duration of the cardiac action potential has a periodic dynamic, they are considered to be pathological rhythms related to the appearance of ventricular fibrillation. Therefore, the study of the dynamic of the APD is aimed to prevent some serious cardiac pathologies such as fibrillation or sudden cardiac death following ventricular arrhythmias.