  Medical practice in the intensive care unit is based on the supposition that
physiological systems such as the human glucose-insulin system are predictable.
We demonstrate that delay within the glucose-insulin system can induce
sustained temporal chaos, rendering the system unpredictable. Specifically, we
exhibit such chaos for the Ultradian glucose-insulin model. This
well-validated, finite-dimensional model represents feedback delay as a
three-stage filter. Using the theory of rank one maps from smooth dynamical
systems, we precisely explain the nature of the resulting delay-induced
uncertainty (DIU). We develop a recipe one may use to diagnose DIU in a general
oscillatory dynamical system. For infinite-dimensional delay systems, no analog
of the theory of rank one maps exists. Nevertheless, we show that the geometric
principles encoded in our DIU recipe apply to such systems by exhibiting
sustained temporal chaos for a linear shear flow. Our results are potentially
broadly applicable because delay is ubiquitous throughout mathematical
physiology.
