Abstract:Since there exist coexisted stable attractors with different attracting basins, when the coupling of different scales involves the vector field, complicated dynamical behaviors can be observed in the multi-stable system with wide science and engineering background. The paper aims at the influence of the coexisted stable attractors on property of the bursting oscillations as well as the mechanism in the multi-stable system. Based on a classical Hatley oscillator, by introducing external periodic excitation, when the frequency is far less than the natural frequency, different forms of bursting oscillations can be obtained with the variation of parameters. Regarding the whole exciting term as a slow-varying parameter, we can derive the equilibrium branches and their bifurcations with the variation of slow-varying parameter by the stability analysis of the equilibrium point of the generalized autonomous fast subsystem. It is found that there exist several situations with different coexisted stable attractors. Here we consider three typical cases, where the characteristics of the coexisted attractors and the influence of their attracting basins on the bifurcation is presented. Accordingly, the bifurcations at the transitions between the quiescent states and spiking states are obtained, which can be used to explore the mechanism of the bursting oscillations. It is pointed out that when the trajectory passes across the attracting basins of the coexisted attractors in turn, the associated attractors may affect the structure of the bursting oscillations, which leads to complicated bursting oscillations. However, when the trajectory only passes across the attracting basins of some of the stable attractors, there may exist coexisted bursting attractors, which correspond to different initial conditions.
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