Bifurcation of dividing surfaces constructed from a pitchfork bifurcation of periodic orbits in a symmetric potential energy surface with a post-transition-state bifurcation

The paper provides a careful computational/qualitative account for a symmetric 2-DoF Hamiltonian with PES V(x,y)=8x^3/3−4x^2+y^2/2+xy^2(y^2−2): it identifies the lower saddle energy (E=−4/3), reports a pitchfork of the lower-saddle periodic-orbit family at E=−0.00056 producing top/bottom families, and describes how periodic-orbit dividing surfaces (PODS) look in various 3D projections, including that the lower-saddle PODS appears ellipsoidal in (x,px,py) with increasing “sharpness,” that the bifurcated PODS lie on opposite sides in projections involving y, and that their ranges/min–max in x and px coincide and track those for the lower-saddle family; it also states the disappearance of the lower-saddle family at E≥0. These claims are consistently documented by figures but lack analytic proofs; in particular, the repeated assertion that the top/bottom PODS “coincide” with the lower-saddle PODS in the (x,px,py) subspace appears to be an observational statement rather than a theorem and, away from a local normal-form neighborhood, is unlikely to be exactly true in general (though it may be visually indistinguishable) . The model, on the other hand, supplies rigorous pieces (invariance of y=0, no-recrossing there, and an exact formula for the (x,px,py)-projection of the lower-saddle PODS with a monotone “sharpness” proxy), but it leaves the pitchfork and bifurcated-family properties as a hypothesis to be checked numerically. Hence, both are incomplete: the paper is empirically persuasive but not proved; the model is analytically sound where developed but conditional on the pitchfork and without completing several proofs.