On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation

The slow-fast effect in a vector field with a codimension-two double Hopf bifurcation at the origin is investigated in the paper. To explore the typical evolution of the dynamics, the universal unfolding of the normal form of the vector field is taken into consideration. When the parametric excitation is introduced, the frequency of which is far less than the two natural frequencies, slow-fast behaviors may appear. Regarding the whole parametric excitation term as a slow-varying parameter, the equilibrium branches and the bifurcations of the generalized autonomous fast subsystem can be derived. With the variation of the exciting amplitude, different types of bursting oscillations caused by the coupling of two scales in frequency domain may appear, the mechanism of which are obtained by employing the overlap of the transformed phase portraits and the equilibrium branches as well as the bifurcations of the fast subsystem. It is found that, for relatively small exciting amplitude, no bifurcation of the fast subsystem occurs, and the system behaves in quasi-periodic oscillations. With the increase of the exciting amplitude, different types of bifurcations may take place, leading to the single-mode bursting oscillations, which may evolve to mixed-mode bursting oscillations. Symmetric breaking bifurcations result in the transitions between a symmetric attractor and two co-existed asymmetric attractors in the system. Furthermore, for the mixed-mode bursting oscillations, the trajectory may alternate between the single-mode oscillations and the mixed-mode oscillations via bifurcations of the equilibrium branches, causing the synchronization and non-synchronization between different state variables.