Topological bifurcations in a mean-field game

TL;DR

This work develops a reduced-order, four-dimensional Hamiltonian BVP for a finite-horizon mean-field game by constraining agent dynamics to the first two moments, the mean $X(t)$ and variance $S(t)$. It leverages phase-space geometry near ergodic equilibria (notably saddle$ imes$saddle and saddle$ imes$center types) and invariant manifolds of unstable periodic orbits to explain the topological structure and multiplicity of solution branches as the horizon $T$ grows. The authors validate the reduced model against full-order MFG PDEs, showing consistent topology (e.g., rotations in the ergodic phase) and density evolution features such as bimodal peaks. The approach highlights a topological bottleneck mechanism that governs transitions between initial and final conditions and suggests natural extensions to higher-dimensional ROMs and richer dynamical regimes for mean-field control and transport problems.

Abstract

Mean-field games (MFG) provide a statistical physics inspired modeling framework for decision making in large-populations of strategic, non-cooperative agents. Mathematically, these systems consist of a forward-backward in time system of two coupled nonlinear partial differential equations (PDEs), namely the Fokker-Plank and the Hamilton-Jacobi-Bellman 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 4D two-point boundary value problem obtained by restricting the controlled dynamics to 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 impart 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.

Figures (16)

• Figure 1: Potential energy surface in (Left) $saddle\times saddle$ and (Right) $saddle\times center$ case.
• Figure 2: Phase portrait of the nonlinear Hamiltonian ODEs in the neighborhood of a $saddle\times saddle$ equilibrium, projected on the (Left) $q_1-p_1$ plane, and (Right) $q_2-p_2$ plane. Also shown are stable (green) and unstable (red) eigenvectors, and the corresponding invariant manifolds.
• Figure 3: The contours of potential energy $V(q_1,q_2)$ (black) separate the allowed (white) and forbidden (green) regions in the configuration space $(q_1,q_2)$ at total energy levels (left) $E_1=E_{eq}$, (middle) $E_2>E_{eq}$, and (right) $E_3>E_2$. The bottleneck around the equilibrium point (red circle) opens for $E>E_{eq}$
• Figure 4: Phase space geometry in the $saddle\times center$ case. Projection of region $\mathcal{R}$ and the trajectories of the linearized Hamiltonian equations on the (left) $\zeta-\eta$ plane, and (right) $\rho_1-\rho_2$ plane. The periodic orbit (purple) projects to the origin on the former, and to a circle of radius $\rho^*$ on the latter. All other trajectories travel on cylinders, and their projections are hyperbolas and circles on the two planes, respectively. The trajectories on cylinders with radius $\rho<\rho^*$ (blue) transit between the bounding spheres, while those on cylinders with $\rho>\rho^*$ (yellow) return back to the originating bounding spheres. Trajectories on the red and green cylinders are asymptotic to the periodic orbit in negative and positive time, respectively.
• Figure 5: The 3D phase space geometry of the nonlinear Hamiltonian ODEs (restricted to a fixed energy level) near a $saddle\times center$ equilibrium point. The energy level $E$ is slightly above that of the equilibrium. The tube-shaped stable (green) and unstable (red) manifolds of the periodic orbit form barriers to transport in this system. Analogous to the linear picture of Fig. \ref{['fig:R_etazeta']}, the trajectories starting inside (blue) the stable tube transit across, while those starting outside (yellow) the tube do not. The transiting trajectories that stay (approximately) on the tubes (purple) are referred to as 'asymptotic' in the main text.