Up to now, most of the work related to the slow-fast dynamical systems are based on the low dimensional systems with only codimension-one bifurcations at the transitions between the quiescent states and the spiking states, while there usually only exists one slow variable in the system. Since the coupling effect of two scales on the behaviors of high dimensional systems with two or more slow variables as well as high co-dimensional bifurcations lacks exploration, in this paper, we investigate the slow-fast effect on a 6-dimension system, which can be regarded as the coupling of two subsystems. The fast subsystem is expressed as the normal form of the vector field with a double Hopf bifurcation at the origin, while the slow-varying parameter in the fast subsystem is controlled by the slow subsystem. For the fast subsystem, the trajectory is projected onto two independent sub-planes for better observation, and the movement of bursting oscillators can be synthesized with information from each dimension. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. It is found that, Hopf bifurcation sets may cause the fast subsystem to transform between the single mode and the mixed-mode or between different single modes, leading to bursting oscillations in certain sub-planes or the whole four dimensions of the fast subsystem. Furthermore, interesting phenomena may be caught, such as the trajectory jumping in the process of oscillation as well as transforming between different oscillation modes of the fast subsystem by the way of jump as a result of the fold bifurcation set.

到目前为止，大多数与慢-快动力系统相关的工作都是基于低维系统，在静止状态和尖峰状态之间的转换处只有一个共维一分岔，而通常只有一个慢变量在系统。由于两个尺度对具有两个或多个慢变量以及高共维分岔的高维系统行为的耦合效应缺乏探索，在本文中，我们研究了对6维系统的慢-快效应，其中可以看作是两个子系统的耦合。快子系统表示为原点具有双Hopf分岔的向量场的范式，而快子系统中的慢变参数由慢子系统控制。对于快速子系统，为了更好地观察，轨迹被投影到两个独立的子平面上，并且可以将爆裂振荡器的运动与来自各个维度的信息进行合成。在子平面中，通过使用变换的相图和平衡分支以及分岔的重叠，推导出突发振荡的机制。研究发现，Hopf分岔集可能导致快速子系统在单模和混合模式之间或不同单模之间转换，导致快速子系统的某些子平面或整个四个维度出现突发振荡。此外，可能会捕捉到有趣的现象，