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

GUSTAVO REVEL; DIEGO M. ALONSO; JORGE L. MOIOLA

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS

SIAM PUBLICATIONS

Lugar: Philadelphia-USA; Año: 2015 vol. 14 p. 1130 - 1164

1536-0040

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 (R1:2), 1:1 strong resonances of period twocycles (R(2)1:1), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1,−1,−1), where the bifurcation curves FF, FNS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when FF interacts with GPD 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.