A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

Abstract

In this paper a degenerate case of a 2:3 resonant Hopf--Hopf bifurcation is studied. This codimension-four 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 (FF) bubble interacting with curves of fold-Neimark--Sacker ($FNS$), generalized period-doubling (GPD), 1:2 strong resonances ($R_{1:2}$), 1:1 strong resonances of period-two cycles ($R_{1:1}^{(2)}$), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1,-1,-1), where the bifurcation curves $FF$, $FNS$, $R_{1:2}$, and CH interact, are detected. A second pair of codimension-three points appears when FF interacts with $GPD$ and $R_{1:1}^{(2)}$ (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.