Bursting is a dynamical behavior that has been widely observed in biological, chemical, and physical systems. It is well-known that the bursting behavior can occur in systems exhibiting distinct fast and slow time scales. Here, we show that bursting can happen in gene regulatory network systems without distinct fast and slow time scales. We perform bifurcation analyses to unravel the mechanisms underlying bursting behaviors in this model. We demonstrate that the bursting behavior is originated from a secondary Hopf bifurcation of a limit cycle, and terminated at a saddle-node bifurcation on an invariant circle. During the bursting cycle, the system evolves from the vicinity of a ghost point due to a disappeared stable fixed point to an unstable focus, then from the unstable focus to an unstable limit cycle, and finally from the unstable limit cycle back to the vicinity of the ghost point again. Our study provides a new mechanism for bursting dynamics in complex systems.

爆裂是一种在生物、化学和物理系统中被广泛观察到的动力学行为。众所周知，爆发行为可能发生在表现出明显快慢时间尺度的系统中。在这里，我们展示了在没有明显快慢时间尺度的基因调控网络系统中爆发可以发生。我们执行分叉分析以揭示该模型中爆发行为的潜在机制。我们证明了爆裂行为源自极限环的次级 Hopf 分岔，并终止于不变圆上的鞍点分岔处。在爆发周期中，系统从稳定不动点消失的鬼点附近演化为不稳定焦点，再从不稳定焦点演化为不稳定极限循环，最终从不稳定的极限环再次回到鬼点附近。我们的研究为复杂系统中的爆发动力学提供了一种新机制。