M. Pernarowski, FAST SUBSYSTEM BIFURCATIONS IN A SLOWLY VARYING LIENARD SYSTEM EXHIBITING BURSTING, SIAM journal on applied mathematics, 54(3), 1994, pp. 814-832
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.