Exact coherent structures and phase space geometry of pre-turbulent 2D active nematic channel flow

The paper explicitly reports that, at non-dimensional activity A ≡ √(α) h^2 / K = 15.5, they find three coexisting attractors—two relative periodic orbits (RPO3a and RPO4a) and a low-dimensional chaotic set (CA)—together with over 40 additional unstable exact coherent structures (ECS) and many heteroclinic connections organizing phase space . The attractors are identified by direct computation (RPO3a, RPO4a listed with zero unstable directions; CA confirmed by a 0–1 test), and families of RPOs/POs are catalogued in tables, with symmetry data under Tk and reflections σx, σy . The global picture is summarized by a directed-graph of ECS and (transverse) heteroclinic edges, and composite transitions are constructed by patching heteroclinics with small instantaneous perturbations—exactly the mechanism the model invokes via the λ-lemma and a reduced graph formalism . However, the paper does not claim and does not provide a mathematically rigorous, parameter-exact proof that there are exactly three attractors or an exhaustive proof-level enumeration of all ECS; instead it documents an extensive but finite computational catalogue and states search heuristics and limits, consistent with a careful computational study rather than a theorem . The model’s solution correctly highlights that fully rigorous certification (e.g., exact enumeration of all attractors/ECS and guaranteed heteroclinic intersections for the PDE at this parameter) is beyond current theory and proposes conditional theorems and a validation protocol. Given this alignment—and the paper’s explicit computational scope—the underlying fully rigorous problem is likely open as of the stated cutoff.