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

G. REVEL; D. M. ALONSO; J. L. MOIOLA

PHYSICA D - NONLINEAR PHENOMENA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2013 vol. 247 p. 40 - 53

0167-2789

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 semiglobal 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.