Basis for finding exact coherent states

Resolvent-based optimization for approximating the statistics of a chaotic Lorenz system

We propose a framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional approaches relying on identifying all short, fundamental unstable periodic orbits to compute ergodic averages via cycle expansion are computationally prohibitive for high-dimensional fluid systems. Our framework stems from the observation in Lasagna [D. Lasagna, Phys. Rev. E 102, 052220 (2020)2470-004510.1103/PhysRevE.102.052220] that a single unstable periodic orbit with a period sufficiently long to span a large fraction of the attractor captures the statistical properties of chaotic trajectories. Given the difficulty of identifying unstable periodic orbits for high-dimensional fluid systems, approximate trajectories residing in a low-dimensional subspace are instead constructed using resolvent modes, which inherently capture the temporal periodicity of unstable periodic orbits. The amplitude coefficients of these modes are adjusted iteratively with gradient-based optimization to minimize the violation of the projected governing equations, producing trajectories that approximate, rather than exactly solve, the system dynamics. An attempt at utilizing this framework on a chaotic system is made here on the Lorenz 1963 equations, where resolvent analysis enables an exact dimensionality reduction from three to two dimensions. Key observables averaged over these trajectories produced by the approach as well as probability distributions and spectra rapidly converge to values obtained from long chaotic simulations, even with a limited number of iterations. This indicates that exact solutions may not be necessary to approximate the system's statistical behavior, as the trajectories obtained from partial optimization provide a sufficient "sketch" of the attractor in state space.

Homoclinic bifurcation and switching of edge state in plane Couette flow

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.