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Mean-field games (MFGs) provide a statistical physics-inspired modelling framework for decision-making in large populations of strategic, non-cooperative agents. Mathematically, these systems consist of a forwards–backwards in time-system of two coupled nonlinear partial differential equations (PDEs), namely, the Fokker–Plank (FP) and the Hamilton–Jacobi–Bellman (HJB) equations, governing the agent state and control distribution, respectively. In this work, we study a finite-time MFG with a rich global bifurcation structure using a reduced-order model (ROM). The ROM is a four-dimensional (4D) two-point boundary value problem (BVP) obtained by restricting the controlled dynamics to the first two moments of the agent state distribution, i.e. the mean and the variance. Phase space analysis of the ROM reveals that the invariant manifolds of periodic orbits around the so-called ‘ergodic MFG equilibrium’ play a crucial role in determining the bifurcation diagram and imparting a topological signature to various solution branches. We show a qualitative agreement of these results with numerical solutions of the full-order MFG PDE system.