In many applied settings, the chemical Langevin equation and linear noise approximation are used in the simulation and data analysis of stochastic reaction networks. With the goal of exploring the sensitivities of reaction network paths to their initial conditions, we subject these modelling techniques to the analysis of random dynamical systems and stochastic flows of diffeomorphisms respectively. After introducing this perspective to stochastic reaction networks in general, we turn our attention to the Brusselator: a two dimensional stochastic reaction network whose paths, when noise is neglected, exhibits a Hopf bifurcation. Studying both Lyapunov exponents, as well as their finite time counterparts, provides two new insights. Firstly, the Brusselator, when modelled by the chemical Langevin equation, exhibits a global synchronisation property of paths of similar noise realisations; secondly, contrary to statistical accuracy in the distributions of concentrations of reactants, the linear noise approximation can fail to capture the finite time dynamical properties of paths of the chemical Langevin equation. In doing so, we explore the notions of dynamical bifurcation and quasi-ergodicity.