Computing Hopf Bifurcations. II: Three Examples From Neurophysiology

In [Guckenheimer, Myers, and Sturmfels, SIAM J. Numer. Anal., 34 (1997)] we present algorithms for detecting Hopf bifurcations in two-parameter families of vector fields based on classical algebraic constructions. In addition to their utility as augmented systems for use with standard Newton-type continuation methods, they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension. The present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology. Implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for Hopf pathfollowing that appear in the literature.