Nonlinear Dynamics: Coherent Structures

Unstable periodic orbits (UPOs) are thought to form the backbone of chaos in driven dissipative nonlinear systems. As exact, non-chaotic, time-periodic solutions of the governing equations, they offer a promising framework for describing and controlling features of transitional turbulence in the fully nonlinear regime. However, identifying UPOs remains a major challenge due to (a) the chaotic nature of the Navier–Stokes equations and (b) the high dimensionality of the state space.

We present a method that addresses both issues by combining recently developed variational convergence algorithms—designed to circumvent time-marching and thereby ‘tame’ chaos—with data-driven nonlinear dimensionality reduction. This yields a convergence algorithm that directly operates in a reduced latent space, in which the search for UPOs becomes more tractable. The approach exploits the tendency of dissipative systems to evolve onto a low-dimensional attractor embedded in the high-dimensional state space. We demonstrate the successful convergence of UPOs in the two-dimensional Navier–Stokes equations.

Funding acknowledgement

This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant No. 865677).