The main purpose of the paper is to display the relaxation oscillations, known as the bursting phenomena, for the coupled oscillators with periodic excitation with an order gap between the exciting frequency and the natural frequency. For the case when the exciting frequency is much smaller than the natural frequency, different types of bursting oscillations such as fold/fold, Hopf/Hopf bursting oscillations can be observed. By regarding the whole exciting term as a slow-varying parameter on the fact that the exciting term changes on a much smaller time scale, bifurcations sets of the generalized autonomous system is derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two cases with typical bifurcation patterns are focused on as examples to explore the dynamical evolution with the variation of the amplitude of the external excitation. Bursting oscillations which alternate between quiescent states (QSs) and repetitive spiking states (SPs) can be obtained, the mechanism of which is presented by introducing the transformed phase portraits overlapping with the bifurcation diagrams of the generalized autonomous system. It is found that not only the forms of QSs and SPs, but also the bifurcations at the transition points between QSs and SPs, may influence the structures of bursting attractors. Furthermore, the amplitudes and the frequencies related to SPs may depend on the bifurcation patterns from the quiescent sates.
By introducing the periodic parameter-switching signal to the Lorenz oscillator, a switched dynamic model is established. In order to investigate the mechanism of the behaviors of the whole system, bifurcation sets of the subsystems are derived and the Poincaré map of the switched system is defined by suitable local sections and local maps. Under certain parameter conditions, symmetric periodic oscillations may be observed. With the variation of parameter, the symmetry-breaking bifurcation mechanisms of the symmetric periodic oscillations can be understood by calculating the associated maximal Lyapunov exponent and Floquet multiplies. Meanwhile, the parameter values of the related local bifurcations, such as saddle-node, pitchfork and period-doubling bifurcations are calculated based on the Floquet multiplies.
