Topological bifurcations in a mean-field game

The paper establishes, via phase-space geometry and numerical continuation, that in the saddle×center regime the Hamiltonian BVP exhibits multiple branches B_n, with (i) multiplicity for sufficiently large horizons (e.g., T > 5.5 in their computations), (ii) along the asymptotic branches B2–B11 the energy increases monotonically with T, and (iii) each branch has fixed topology measured by half-rotations, with an explicit relation τ_erg/t_p = 0.5 n, all organized by codimension-1 tube manifolds of a Lyapunov periodic orbit. These points are documented in their figures and discussion of tube dynamics and the E–T bifurcation diagram, and in their explanation that only trajectories inside the stable/unstable tubes transit and that branch origins correspond to the lowest energy achieving a given number of half-rotations (see the tube/barrier explanation and existence of Lyapunov orbits, and the branch properties in the E–T diagram and its commentary . The candidate model solution provides a complementary, theory-driven sketch: it invokes the Lyapunov–(sub)center theorem, the existence of 2D stable/unstable tubes in the 3D energy surface, a time-of-flight decomposition T(E) = τ_in(E) + (n/2) t_p(E) + τ_out(E), and the implicit function theorem under a non-isochrony condition t_p'(E) ≠ 0 to produce smooth branches B_n with E(T) monotone for large T. This analytical scaffold aligns with the paper’s observations and offers a principled explanation of monotonicity and branch organization. Minor differences are mainly notational (indexing of branches vs. half-rotations) and scope (the paper does not attempt a formal proof nor assert existence for all n; the model’s proof sketch assumes standard hypotheses like non-isochrony and smooth sections). Net: the empirical/geometric argument and the theoretical sketch agree on substance.