It’s not true that neighboring initial conditions will converge to the fixed state after long time. As we saw from the Jupiter problem, there are some regions where the initial conditions for chaotic and periodic trajectories lies infinitesimally close but lead to different results. The chaotic initial conditions will not make the system converge to certain state. Therefore, there is no attractor in this 3-body problem.
Stable Hamiltonian systems can withstand a minor perturbation in the process and the system will still end up where it’s supposed to.
Unstable Hamiltonian system will diverge if perturbed in the process and the system will not come to the original equilibrium regardless of whether the system becomes chaotic or not.