Conjectured behavior of solutions in the gap between P12 and the B2 branch

Determine whether solutions of the 4D Hamiltonian boundary value problem defined by the equations of motion (Eq. \ref{eq:Ham_ode}) with boundary conditions (Eq. \ref{eq:ic_fc_ham}) in the saddle×center parameter regime (\sigma=1, \mu=2, g=4, \alpha=3, h=0, \epsilon=0.05; initial/final conditions q1(0)=-10, q2(0)=4.5, q1(T)=10, q2(T)=4.5) that lie between the termination point P12 of branch B1 and the lowest-energy point on branch B2 behave analogously to the solutions on the segment from P23 to the lowest-energy point on branch B3, specifically having the topology of branch B3 while traveling inside (rather than on) the stable/unstable invariant manifolds (tubes) of the hyperbolic periodic orbit.

Background

In the saddle×center case, the authors analyze the phase-space geometry near a hyperbolic periodic orbit whose stable and unstable manifolds form tube-like barriers to transport. They compute multiple solution branches B1, B2, B3, … of the 4D Hamiltonian boundary value problem (BVP), where each branch is topologically classified by the number of half-rotations a trajectory performs around the tube during the ergodic phase.

They observe that the B1 branch terminates at a point P12 because the 1D invariant manifolds of the equilibrium do not intersect the final-condition line, implying a lower bound on the tube radius for feasible trajectories. Numerical continuation failed to produce solutions in the interval between P12 and the lowest-energy point on B2.

Based on the behavior of related segments (e.g., from P23 to the lowest-energy point on B3), the authors conjecture that the missing solutions in this gap share the topology of B3 but travel inside the tubes rather than on them, with tube size shrinking as energy decreases and trajectories approaching the tubes until the branch B3 originates.