Invariant manifolds in Hamiltonian systems with applications to the Earth-Moon system

The uploaded paper (2105.07016.pdf) numerically documents an order–chaos–order transition on a Moon-centered Poincaré section Σ={xM<x<xL2, y=0, ẏ>0}, identifies the two codimension‑1 events for the direct lunar period‑1 fixed point (loss of stability at C≈3.18451 and a saddle–node at C≈3.18266), shows Lyapunov-manifold “intersect-and-break” geometry, and introduces a multiplicative transit time τ=|tftb| whose minima lie along Γs(L)∩Γu(L) while maxima cluster near selected sticky UPOs P7_I and P8_III. All of these match the model’s core claims and specific values (e.g., scenarios I–II–III and Cbif’s) precisely, as seen in the paper’s definitions of Σ and Hill stability, the three-scenario phase portraits, the bifurcation analysis, the geometrical evolution of Γ(L), the intersect-and-break mechanism, the definition and behavior of τ, and escape statistics via a mean escape measure νi (e.g., “black lakes”/fast channels along Γs∩Γu). The model complements the paper with standard symplectic-map theory (KAM, bifurcation normal forms, λ‑lemma/Smale–Birkhoff, lobe dynamics) to justify the observations. One nuance: the model asserts τ can be made “arbitrarily small” near transverse homoclinic points; the paper only demonstrates observed minima, and a rigorous statement should acknowledge a positive lower bound set by finite flight times on the flow. Otherwise, there is strong agreement, with the model adding theoretical underpinnings to the paper’s numerical evidence (e.g., density of W(L) in the chaotic sea and measure-zero survivors stated in the paper). Key correspondences: Σ and its constants (), I–II–III scenarios with C windows and bifurcations C1_bif≈3.18451 and C2_bif≈3.18266 (), Lyapunov-manifold spread and relation to KAM destruction/new island (), τ definition/behavior and stickiness near P7_I and P8_III (), intersect-and-break and homoclinic/heteroclinic connections (), Hill stability/geometry (), and ensemble escape-measure channels along Γs∩Γu ().