On the influence of natural frequency on the double Hopf bifurcation system forced by excitations with a low frequency

Abstract

This paper mainly focusses on the influence of natural frequency on a double Hopf bifurcation system with two-time scale. While the attractors may vary as the excitation amplitude expands, we try to discover all the characteristic of influence by gradually increasing the excitation amplitude. Based on the grasped mechanism of oscillators as well as comparative research concentrating at different values of natural frequency, some regular conclusions can be reached. When the excitation amplitude is small, only oscillators of single mode exist, while the natural frequency responsible for the sub-plane with no oscillation makes no difference to the one with oscillations. With the amplitude increasing, the system may switch between oscillations of different modes. The natural frequency may affect the frequency of bursting oscillation in its own sub-plane and the time that the system switches from single-mode oscillation to mixed-mode oscillations. For relatively large amplitudes, oscillations with mixed modes occupy the whole movements. The natural frequency corresponding to one sub-plane may affect the amplitude of bursting oscillation or the time that the system alternates between the spiking state and the quiescent state as well as the routes that the trajectory goes through in the other sub-plane.