3. Symmetries of plane Couette flow

s₁ :  [u,v,w](x,y,z) -> [u,v,-w](x+L_x/2, y, -z)

s₂ :  [u,v,w](x,y,z) -> [-u,-v,-w](-x+L_x/2, -y,- z+L_z/2)

By “equivariant” we mean that if u(x,t) is a solution, so are s₁u(x,t), s₂u(x,t), and s₁s₂u(x,t); the four solutions are physically equivalent. (Click here to see examples of such symmetry-related solutions).

We refer to the space of velocity fields left invariant under the symmetry group S = {1, s₁, s₂, s₁s₂} as the S-invariant subspace. All the exact invariant solutions (equilibria and their stable/unstable manifolds, periodic orbits) shown in what follows belong to this subspace.