Etude de systemes dynamiques chaotiques. Analyse et applications

Abstract

This thesis is based on the rigorous application of the theory of nonlinear dynamics and chaos to cardiac electrophysiology and more precisely to the control of cardiac alternans with a control signal constructed from measurements of the action potential duration (APD). A first axis of this work consists in a new approach by adapted control methods in order to delete an asymptotically stable periodic orbit undesired. The second axis concerns the development of control methods to achieve not only the suppression of unwanted stable periodic dynamics (such as cardiac alternation leading to ventricular fibrillation and sudden death) but also to control other chaotic dynamics than heart dynamics. The third axis focuses on the development of a mathematical model of the APD with memory to obtain a new control parameter, and a spectrum of new bistabilities between some cardiac rhythms. The practical benefits of this theoretical mathematical work is to achieve control of cardiac alternation with the prospect of improving implantable electronic devices designed to suppress cardiac alternation.