  Biochemical reactions with oscillatory behavior play an essential role in
synthetic biology at the microscopic scale. Although a robust stability theory
for deterministic chemical oscillators in the macroscopic limit exists, the
dynamical stability of stochastic oscillators is an object of ongoing research.
The Chemical Master Equation along with kinetic Monte Carlo simulations
constitute the most accurate approach to modeling microscopic systems. However,
because of the challenges of solving the fully probabilistic model, most
studies in stability analysis have focused on the description of externally
disturbed oscillators. Here we apply the Maximum Entropy Principle as closure
criterion for moment equations of oscillatory networks and perform the
stability analysis of the internally disturbed Brusselator network.
Particularly, we discuss the effects of kinetic and size parameters on the
dynamics of this stochastic oscillatory system with intrinsic noise. Our
numerical experiments reveal that changes in kinetic parameters lead to
phenomenological and dynamical Hopf bifurcations, while reduced system sizes in
the oscillatory region can reverse the stochastic Hopf dynamical bifurcations
at the ensemble level. This is a unique feature of the stochastic dynamics of
oscillatory systems, with an unknown parallel in the macroscopic limit.
