Applying the Conley index to fast-slow systems with one slow variable and an attractor

Chay and Keizer [3] created a five-dimensional model of bursting activity in pancreatic beta-cells which was subsequently reduced to a three-dimensional model by Chay [2]. Kinney has used the Conley index to show that the three-dimensional model has a nonempty attractor [11, pages 451-472]. This paper is intended to provide an introduction to the Conley index by showing how it can be applied to extend these results to prove the existence of a periodic orbit for the three-dimensional model, the existence of a nonempty attractor for the five-dimensional model and the existence of a periodic orbit for the five-dimensional model.