The Phase Space Structure in the vicinity of vertical Lyapunov orbits around L1,2 in a barred galaxy model

The paper establishes, at Ej = −0.165 on the 4D section z=0, pz>0, the five-type classification (double-crescents, clouds, quasi-periodic tori, sticky tori, and toroidal zones), gives precise perturbation windows for double-crescents, lists the relevant periodic-orbit initial conditions, and defines/measures diffusion via Vd(t), Td, Vmax, ud. The candidate solution reproduces the same setup, windows, mechanisms, morphology, and diffusion-speed values (e.g., ud ≈ 0.624 for double-crescents; 0.14832 for clouds; 0.001034 for a sticky torus; 0.0088 for a toroidal zone), and explains the role of sao/vLPO4,5 tori and unstable manifolds in near-identical terms to the paper. Minor additions (e.g., generic remarks about transverse manifold intersections and an explicit formula for the 4D hypersphere volume) go slightly beyond what is stated but do not conflict with the paper. Overall, both are consistent in claims and evidence, with essentially the same argument and conclusions supported by the paper’s figures and diagnostics.