Conjecture on approximate invariant manifolds and chaotic dynamics in the three-body problem

Let ${\cal M}({\cal E})=\{({\rm r},{\rm G},{\rm g}):{\rm E}({\rm r},{\rm G},{\rm g})={\cal E}\}$ be the lifted level manifolds of the Euler integral, and let ${\cal M}_0=\{({\rm r},{\rm G},{\rm g}):{\rm E}({\rm r},{\rm G},{\rm g})={\rm r}\}$. Consider parameters and a phase-space region where the scale conditions

$$\|{\rm K}_{\rm C}\|\gg\alpha\|{\rm U}\|\gg\|\widetilde{\rm K}_{\rm C}\|$$

hold. Approximate-invariant-manifold and chaos conjecture. The manifolds ${\cal M}({\cal E})$ are approximate invariant manifolds for the Hamiltonian ${\rm H}_{\rm 3b,{\rm C}}$, at least while ${\rm r}$ remains in a fixed region, and the coupling between ${\rm K}_{\rm C}$ and ${\rm U}$, the disturbing term $\widetilde{\rm K}_{\rm C}$, and the remainder ${\rm O}_2$ generate chaotic dynamics near ${\cal M}_0$. The conjecture is motivated by the phase portraits and numerical explorations; the supplied text gives no resolution, and the later assertion is explicitly described as the weaker statement supported numerically.