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Bi-directional (meta-homoclinic) sequences and their stability

Investigate the existence of meta-homoclinic or bi-directional sequences for which trajectories converge to a heteroclinic or homoclinic network in both forward and backward time, and determine whether such sequences are fragmentarily asymptotically stable.

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Background

The notion of fragmentary asymptotic stability concerns forward-time convergence. The authors ask whether there exist sequences permitting convergence in both time directions and, if so, whether such sequences attract a positive-measure set.

This would extend the taxonomy of stable itineraries and connect to concepts of higher-depth heteroclinic structure.

References

We conclude by discussing some open questions, whose answers are beyond the scope of this note. Finally, can we have a meta-homoclinic (or bi-directional) heteroclinic sequence in the sense that there are trajectories converging to a network in forward and backward time, and if so, are they f.a.s.?

— How many points converge to a heteroclinic network in an aperiodic way? (2410.11383 - Bick et al., 15 Oct 2024) in Section 6 (Discussion)