Exploiting invariant manifolds for optimal control in active hydrodynamic systems*

Active hydrodynamics equations give rise to deterministic nonlinear dynamical systems evolving in infinite dimensional phase spaces. The dominant flow structures are understood in terms of Exact Coherent Structures (ECS). An ECS is (generically unstable) stationary, periodic, quasiperiodic, or traveling wave solution of the hydrodynamic equations. These ECSs are interconnected by invariant manifolds, which act as dynamical pathways. A finite set of ECS, together with their invariant manifolds, constitutes a reduced-order but exact characterization of the global phase space. Though each ECS is non-turbulent, this representation has been shown to be adequate for describing high Reynolds number turbulent flows of passive fluids, which appear as chaotic trajectories meandering through the phase space and visiting the neighborhoods of different ECS in a recurring fashion. 

    We provide evidence that the ECS and their invariant manifolds also form an organizing template for the complicated spatiotemporal motion of active fluid turbulence. For active nematic channel flow, we obtain a reduced-order representation of the phase space in terms of a directed graph, in which ECS are nodes, and dynamical connections are edges. This representation uncovers nontrivial relationships in phase space, which can be exploited to induce desired transitions between disparate spatiotemporal states using minimal external control input. Examples include switching between the left and right flowing laminar states, or between a stationary state and an oscillatory vortex lattice. Our results lay the groundwork for a systematic means of understanding and controlling active fluids in the moderate- to high-activity regime.