BIFURCATIONS IN FLAT-TOPPED MAPS AND THE CONTROL OF CARDIAC CHAOS

Abstract

A recent experiment in cardiac physiology [Garfinkel et al., 1992] controlled irregular interbeat intervals by delivering a stimulus that was meant to perturb the trajectory to the stable manifold of an unstable fixed point. This paper offers an alternative mechanism for regularizing cardiac rhythms. In analogy with the operation of commercial cardiac pacemakers, we analyze a mechanism for chaos control in which a stimulus is delivered at a fixed delay, provided the preceding interbeat interval lies in a well-defined range. This mechanism can be characterized by maps with flat segments that we analyze using standard methods of symbolic dynamics. This mechanism is compared with a control scheme in which a parameter is perturbed repeatedly to stabilize an unstable fixed point.