Asymptotic behavior of Rt sets of reachable states

Characterize, for a given hybrid gene regulatory network ๐“, the asymptotic behavior of the sets Rt = { h(t) | h โˆˆ H๐“ } where H๐“ denotes the set of all possible hybrid trajectories (including indeterministic choices). Specifically, determine for large t the probability that Rt approaches a zero-set and quantify how fast this approach occurs.

Background

The authors consider the set H๐“ of all trajectories in an HGRN and the time-slice sets Rt collecting the states reached at time t by any admissible trajectory. Understanding how Rt evolves is linked to questions of chaos, indeterminism, and reachability in HGRNs.

They ask for a probabilistic and quantitative characterization of whether and how quickly Rt collapses toward a zero-set as time increases.

References

Further open questions are: 3.) For $H_\mathcal N$ the set of all possible trajectories in $\mathcal N$, how do the sets $R_t:={h(t)\mid h\in H_\mathcal N}$ behave for large $t$? With what probability and how fast do they approach a zero-set?

— On Hybrid Gene Regulatory Networks  (2404.16197 - Wurm et al., 2024) in Section "Conclusion and Further Questions", Item 3