Conjectured absence of f.a.s. aperiodic sequences in systems with infinite switching

Prove or refute that in heteroclinic networks exhibiting infinite switching due to the presence of equilibria with non-real eigenvalues, no aperiodic sequence is fragmentarily asymptotically stable.

Background

In systems with infinite switching—often enabled by complex eigenvalues at equilibria—any sequence may be realized by trajectories, frequently on invariant sets of measure zero. The authors argue that the analysis in such systems does not suggest qualitative differences in stability across sequences and pose a conjecture of absence of f.a.s. aperiodic sequences.

Resolving this conjecture would clarify the interplay between switching phenomena and positive-measure attraction, and delineate where aperiodic f.a.s. dynamics can or cannot occur.