In this paper, we deal with a Finsler–Randers (FR) cosmology in the framework of particle creation mechanism. The cosmological history of the model is studied by finding all the critical points and analysing their local stability.We study the behaviour of all critical points of the model when they are non-hyperbolic in nature using the centre manifold theory. The perseverance of the equilibrium points are illustrated by showing the vector field locally near the equilibrium point. The possible bifurcation scenarios are discussed in detail with the help of local bifurcation diagrams for each of the critical points. Concluding remarks and the cosmological upshot are also given.