Global invariant manifolds delineating transition and escape dynamics in dissipative systems

The paper explicitly states and numerically demonstrates that, in 2-DOF systems with a single index-1 saddle, the boundary of the transition region at fixed initial energy h is the stable invariant manifold: a cylindrical transition tube around a hyperbolic periodic orbit in the conservative case, and an ellipsoid given by the intersection of the 3D stable manifold of the saddle equilibrium with the initial-energy label in the dissipative case. See the abstract and overview claims (cylinder vs ellipsoid, stable manifolds as separators) and the linear and nonlinear constructions, where the stable manifold at energy h is identified as ∂Th in the conservative case and the transition ellipsoid is a subset of the equilibrium’s stable manifold in the dissipative case . The model solution gives a complementary, theorem-driven justification: Lyapunov Center Theorem and NHIM/stable-manifold geometry in the conservative case, and a Lyapunov-energy/stable-manifold argument in the dissipative case (using dH/dt ≤ 0 and the stable manifold theorem), arriving at exactly the same separating objects and topologies. Minor phrasing differences aside (e.g., the paper sometimes says the ellipsoid 'gives initial conditions for all transit orbits' where it functions as the boundary), the mathematical content and conclusions agree closely, including the no-recrossing/transversality, Floquet multipliers of the periodic orbit, and the cylinder (S^1 × I) vs ellipsoid (S^2) topology .