Existence and measure of trajectories following aperiodic sequences

Determine, for heteroclinic or homoclinic networks, whether for any prescribed aperiodic sequence of connections there exists a trajectory that follows that sequence, and ascertain the Lebesgue measure of initial conditions that follow any given aperiodic sequence.

Background

The paper highlights a gap between stability analyses of cycling dynamics (which focus on periodic or preperiodic itineraries) and switching (which concerns the existence of trajectories that follow arbitrary itineraries). Numerical evidence suggests the possibility of aperiodic convergence to networks, but a rigorous understanding of whether arbitrary aperiodic itineraries are realized and, if so, how many initial conditions follow them, remains to be developed.

The authors introduce a notion of fragmentary asymptotic stability (f.a.s.) for sequences and derive constraints on how many aperiodic f.a.s. sequences can exist. A foundational question is the existence of trajectories realizing aperiodic itineraries and quantifying the measure of such initial conditions.