Numerical semi-global analysis of a 1:2 resonant Hopf–Hopf bifurcation

In this paper, a numerical semi-global analysis of the dynamics near a 1:2 resonant Hopf–Hopf bifurcation on a four-dimensional mathematical model of a simple nonlinear oscillator is performed. The 1:2 resonant Hopf–Hopf bifurcation is a codimension-three singularity denoted by two pairs of purely imaginary eigenvalues with frequency ratio 1:2. A structure involving 1:1 and 1:2 resonant Neimark–Sacker bifurcations is clearly identified. Both resonances are coupled by lower-codimension singularities, such as generalized Hopf and period doubling, cusp points, and cyclic fold curves. A three-parameter semi-global analysis is performed and some of the codimension-two singularities unfolded by the 1:2 resonant Hopf–Hopf bifurcation are identified. Several codimension-three points are also detected. The obtained results can be useful for further theoretical analysis of the corresponding normal form.

► A semi-global analysis of the 1:2 resonant Hopf–Hopf bifurcation is performed. ► Two structures including 1:1 and 1:2 resonant Neimark–Sacker are identified. ► Numerical continuation of bifurcations in three parameters are carried out. ► A partial unfolding of the 1:2 resonant Hopf–Hopf singularity is included. ► Codimension-three singularities are detected near the 1:2 resonant Hopf–Hopf point.