Fast Subsystem Bifurcations in a Slowly Varying Liénard System Exhibiting Bursting

Abstract

A perturbed Lienard differential system is examined using local stability 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 space, the model exhibits square-wave bursting patterns analogous to the bursting electrical activity observed in pancreatic ,$\beta $-cells. Under certain hypotheses, solutions of the cubic model are known to have square-wave patterns. By using the theory developed for the more general Lienard system, each hypothesis is shown to correspond to a curve in parameter space. Together, the curves bound a region in which the model exhibits square-wave bursting patterns. Since the model is simple, the curves that bound this region can all be determined analytically.