FAST SUBSYSTEM BIFURCATIONS IN A SLOWLY VARYING LIENARD SYSTEM EXHIBITING BURSTING

A perturbed Lienard differential system is examined using local stabil ity and Hopf bifurcation analyses, asymptotic techniques, and Melnikov 's method. The results of these analyses are applied to a simple cubic model that exhibits a variety of different oscillatory behaviors for different parameter values. For a bounded region in (fast) parameter s pace, the model exhibits square-wave bursting patterns analogous to th e bursting electrical activity observed in pancreatic beta-cells. Unde r certain hypotheses, solutions of the cubic model are known to have s quare-wave patterns. By using the theory developed for the more genera l Lienard system, each hypothesis is shown to correspond to a curve in parameter space. Together, the curves bound a region in which the mod el exhibits square-wave bursting patterns. Since the model is simple, the curves that bound this region can all be determined analytically.