A Degenerate 2:3 Resonant Hopf-Hopf Bifurcation as Organizing Centre of the Dynamics: Numerical Semi-Global Results

Abstract

In this paper a degenerate case of a 2:3 resonant Hopf–Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two- and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and Neimark– Sacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F) bubble interacting with curves of fold-Neimark–Sacker (FNS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R(2) 1:1), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F, FNS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf–Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.