Impact of unstable fixed points on variational optimisation for fluid systems

Determine the effect of multiple unstable fixed points in the state space of high-dimensional fluid dynamical systems on the variational optimisation procedure that constructs periodic quasi-trajectories by minimising the global residual of the governing equations projected onto a resolvent-based subspace.

Background

In the Lorenz system case study, the authors observe that the variational optimisation produces quasi-trajectories whose residual is small near fixed points regardless of stability, which can cause trajectories to drift toward an unstable fixed point (the origin) more than occurs in chaotic simulations. They note that this behaviour is a consequence of how the residual is constructed and minimized.

Extending this observation to fluid problems, the authors highlight that such systems may contain many unstable fixed points scattered throughout state space. They explicitly state that the precise influence of these fixed points on the optimisation outcome is not known and identify it as a topic requiring future investigation.