Interaction between fold and Hopf curves leads to new bifurcation phenomena

Abstract

This paper presents a numerical investigation of the complex phenomena which can occur at an interaction between fold and Hopf curves. In a two-parameter problem, qualitative information about steady state and periodic solutions can be obtained by computing the bifurcation set, consisting of fold curves and curves of Hopf points. This paper studies the evolution of the bifurcation set with respect to a third parameter for two mathematical models, the Brusselator trimolecular reaction scheme and a tubular reactor model. We find that a limit point of a branch of B-points coincides with a cusp point of a fold curve. At such a limit point branches of Hopf curves can disappear or can be detached from the fold curve. Bifurcation of two fold curves at a cusp point and bifurcation of Hopf curves has been detected too.