Degenerate Hopf Bifurcation and Isolas of Periodic Solutions in an Enzyme-Catalyzed Reaction Model

A point of degenerate Hopf bifurcation in an enzyme-catalyzed model is rigorously analyzed by using techniques of singularity theory and interval analysis. The existence of the point is proven by a numerical computation using interval analysis. Singularity theory as developed by M. Golubitsky and D. G. Schaeffer is then used to completely characterize the families of small amplitude periodic solutions that arise for parameters near the generate values. The normal form for the singularity is X5 + 2m0ΛX3 + Λ2X, with Λ the bifurcation parameter, m0 a model parameter. Among the results is the existence of an isolated branch (isola) of periodic solutions. Purely numerical techniques are used to confirm the results: excellent agreement is found between the bifurcation theoretic unfolding and (numerical) continuation results using pseudo-arclength continuation.