Videos accompanying the paper "Symmetry, Invariant Manifolds and Flow Reversals in Active Nematic Turbulence"

Supplementary Information Symmetry, Invariant Manifolds and Flow Reversals in Active Nematic Turbulence Angel Naranjo1, Rumayel Pallock1, Caleb G. Wagner2, and Piyush Grover1 1Mechanical and Materials Engineering, University of Nebraska–Lincoln, Lincoln, NE 68588, USA 2Parsons Corporation, USA CONTENTS I. List and description of movies ..............................................................2 II. Methods ..................................................................................2 1 1. Description of movies Name (.mp4) Description movie1 DPO T1a Ra1.5.mp4 Nematic director and velocity field for the periodic orbit DPOT1aat Ra= 1.5 in the channel. movie2 PO T2a Ra2.25.mp4 Nematic director and velocity field for the periodic orbit POT2aat Ra= 2.25 in the channel. movie3 HRPO T1a Ra1.24.mp4 Nematic director and velocity field for the homocliniclike relative periodic orbit HRPOT1aat Ra= 1.24 in the channel. movie4 HRPO T2a Ra1.24.mp4 Nematic director and velocity field for the homocliniclike relative periodic orbit HRPOT2aat Ra= 1.24 in the channel. movie5 HTPO T1a Ra1.5.mp4 Nematic director and velocity field for the heterocliniclike periodic orbit HTPOT1aat Ra= 1.5 in the channel. movie6 HTPO T1b Ra1.5.mp4 Nematic director and velocity field for the heterocliniclike periodic orbit HTPOT1bat Ra= 1.5 in the channel. movie7 One vortex TW.mp4 Nematic director and velocity field for the one–vortex traveling wave at Ra= 0.851 in the channel. movie8 One vortex standing wave.mp4 Nematic director and velocity field for the one–vortex standing wave at Ra= 0.93 in the channel. movie9 Preturbulent reversals Ra1.5.mp4 Nematic director and velocity field for a preturbulent trajectory at Ra= 1.5 in the channel exhibiting reversals. movie10 T2 4 Ra4.5.mp4 Nematic director and velocity field for the green (Vortex lattice RPO) ECS T2/4 at Ra= 4.5 in the channel. movie11 T2 1 Ra4.5.mp4 Nematic director and velocity field for the blue (homoclinic-like RPO) ECS T2/1 at Ra= 4.5 in the channel. movie12 Shadowing turbulent.mp4 Example of shadowing of an ECS and its unstable manifold in turbulent regime. movie13 Turbulent Three reversals in X.mp4 Nematic director and velocity field for a turbulent trajectory with three reversals of Xin the channel and in the reduced phase space representation. 2. Methods Our calculation framework uses extensively the open toolkit Exact Coherent Structures in Active Matter (ECSAct) which is based on the open-source pseudospectral code Dedalus. 2 2.1. Symmetry and Equivariant Bifurcation Analysis For local equivariant analysis, we employed the symmetry tools provided by the ECSAct code to identify the isotropy subgroups associated with the bifurcating ECSs and their corresponding eigenspaces. In several instances, additional analysis required projecting full time-dependent simulations onto the appropriate invariant subspaces in order to get solutions that are stable within those subspaces and predicted by the equivariant bifurcation theory. For global bifurcation analysis, we needed to compute multiple RPOs to confirm the presence of SNIPER, homoclinic, and heteroclinic bifurcations. Long-period RPOs were particularly challenging, since the associated Newton solver often struggled with very large temporal periods nearby the bifurcation point. To mitigate this difficulty, we projected the solutions onto the invariant subspace in which the relevant RPO or PO is more stable. For example, every global-bifurcation branch associated with the first complex unstable pair of UNI required projecting the trajectories that initiated from the bifurcating ECS onto the subspace invariant under σxτx(L/2) by applying the operator, P=I+σxτx(L/2) 2(1) 2.2. Heteroclinic Connections and Shadowing Verification 2.2.1. Heteroclinic Connections Heteroclinic connections are confirmed using the distance function d(X1(t), XE) = min 0≤`≤L 0≤s≤T kτx(`)X(t)−Xtarget(s)k2< , (2) where X(t) is the trajectory originating from a source ECS along its unstable manifold. In practice, the unstable manifolds were approximated using approximately 100 to 1000 points along the unstable directions. In general, heteroclinic connections were difficult to verify to tolerances much smaller than 10−2, unless the target ECS lies within an attractor contained in a specific invariant subspace. However, if the unstable manifold of the source ECS is one-dimensional, or dominated by a single unstable eigenvalue, and the target ECS is not too unstable, a shooting method can be applied by minimizing the function d2(µ, t, `, s) = kφ(X0+µv, t)−τx(`)Xtarget(s)k2 2,(3) where vis the dominant unstable eigenvector. The corresponding minimization problem, min µ, t, `, s d2(µ, t, `, s),(4) can be solved using Gauss-Newton or Levenberg-Marquardt methods. Using the former, with a good initial guess, we successfully verified the heteroclinic connections DPOT1b−→ RPOT1a,HTPOT1a−→ DRPOT1a, with tolerance as low as 10−8and 10−6, respectively. Although the shooting formulation can be generalized easily for the case where the unstable manifold is effectively multidimensional, we could only verify such connections with a tolerance of 10−2. An adjoint-based method could be used to confirm these connections to a lower tolerance value in the future. 3 2.2.2. Shadowing Although the distance calculation already gives us direct information of shadowing events, we also verified the correspondence of the dynamics between the trajectory and the target ECSs. In calculation of the numerical derivatives ds dt and d` dt, the discontinuities of sand `for a relative periodic orbit uwith period Tand shift `0need to be accounted for. To do so, we use the relation τx(`o)u(T) = u(0). The derivative d` dt also allowed us to verify shadowing of unstable manifolds. By considering a shadowed ECS, we looked for trajectories on its unstable manifold that are shadowed for a nontrivial time interval while satisfying d` dt ≈1. By doing this, we found for example the shadowing of the unstable manifold of σxσyT2/1 in the interval [29064τ, 29094.4τ]. 4