Bursting and complex oscillatory patterns in a gene regulatory network model

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.