Explaining flow patterns by non-existing solutions of the governing equations*

Invariant solutions of the governing equations, such as unstable equilibria and periodic orbits, are believed to serve as elementary building blocks of chaotic fluid flows and to play a major role in the emergence of patterns and coherent flow structures. Close to a saddle-node bifurcation, when two invariant solutions collide and annihilate, the flow behavior can closely resemble that of the solution at the bifurcation point, even though the solution itself does not exist at the studied parameter value. Therefore, patterns and coherent flow structures may emerge as a result of the dynamics feeling a non-existing invariant solution, a phenomenon called the ‘ghost’ of a solution. 

We use recently developed variational methods to formalize the concept of ghosts, follow ghost states in parameter space and demonstrate how even non-existing invariant solutions may control the behaviour of the chaotic flows, including 3D Rayleigh-B\'enard convection.