A global 3d portrait of equilibria and their manifolds

While different equilibria are clearly physically distinct, it is hard to intuit how are they related dynamically from their spatial 3D portraits. The global dynamics of the transitionally turbulent flows is revealed by their state space portraiture.

 

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A family portrait of 25 exact equilibria and traveling waves of plane Couette flow projected onto the translational basis constructed from half-cell shifts τ_x, τ_z, and τ_xz of UB . The dots denote equlibirium points, bowties are traced out by traveling waves.

 

black dot:

laminar state LM

black lines:

LB (red dots) unstable manifold heteroclinic connection to LM

bowties:

traveling waves and their countertraveling partners

blue spirals:

portions of UB unstable manifold.