Unfolding a Point of Degenerate Hopf Bifurcation in an Enzyme-Catalyzed Reaction Model

Abstract

A point of degenerate Hopf bifurcation in an enzyme-catalyzed model previously studied by Doedel [2] is rigorously analyzed by using techniques of singularity theory and interval analysis. A computation using interval analysis proves the existence of a point of degenerate Hopf bifurcation, which is a smooth function of additional parameters in the model system. Singularity theory as developed by Golubitsky and Langford [J. Differential Equations, 3 (1981), pp. 375–415.] and Golubitsky and Schaeffer [Singularities and Groups in Bifurcation Theory I, Springer-Verlag, New York, 1985.] is then used to construct universal unfoldings of the degeneracy, to completely characterize the families of small amplitude periodic solutions that arise for parameters near the degenerate values. Computations using interval analysis are employed in this proof. Excellent agreement is found between the bifurcation theoretic unfolding and (numerical) continuation results using pseudoarclength continuation.