Edge manifold as a Lagrangian coherent structure in a high-dimensional state space

Abstract

Dissipative dynamical systems characterized by two basins of attraction are found in many physical systems, notably in hydrodynamics where laminar and turbulent regimes can coexist. The state space of such systems is structured around a dividing manifold called the edge, which separates trajectories attracted by the laminar state from those reaching the turbulent state. We apply here concepts and tools from Lagrangian data analysis to investigate this edge manifold. This approach is carried out in the state space of autonomous arbitrarily highdimensional dissipative systems, in which the edge manifold is reinterpreted as a Lagrangian coherent structure (LCS). Two different diagnostics, finite-time Lyapunov exponents and Lagrangian descriptors, are used and compared with respect to their ability to identify the edge and their scalability. Their properties are illustrated on several low-order models of subcritical transition of increasing dimension and complexity, as well on wellresolved simulations of the Navier-Stokes equations in the case of plane Couette flow. They allow for a mapping of the global structure of both the state space and the edge manifold based on quantitative information. Both diagnostics can also be used to generate efficient bisection algorithms to approach asymptotic edge states, which outperform classical edge tracking.

Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

“…2018; Blanchard & Sapsis 2019 c ), prediction of dynamical events in a statistical framework (Farazmand & Sapsis 2016), the computation of sensitivities (Donello, Carpenter & Babaee 2020) to edge tracking (Beneitez et al. 2020), leveraging the ability of the OTD modes to follow the linear dynamics even along a chaotic trajectory.…”

Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems

“…Remark 1. Whilst the regularity of the LD metric has already been quantified through first derivatives (see, e.g., [8]), we found it to be too limiting for our purpose, and especially for the problems addressed in Sect. 3.2 and 4.2 where the geography of the resonances, in the "actions" plane, is portrayed.…”

“…As a matter of fact, the computation of those arc-lengths, as function of the coordinates, is able to locate and reveal the geometrical template organising the phase space. These structures include objects such as separatrices of hyperbolic equilibria, the stable and unstable manifolds of hyperbolic orbits, manifolds of normally hyperbolic manifolds, or generalisations such as Lagrangian coherent structures [33,30,31,25,8,1]. The heuristic idea driving the ability of the LD to detect hyperbolic objects, as described in [37,33], is that trajectories that start and evolve close to each other will have similar arc-lengths, whilst those arc-lengths will change "abruptly" when crossing separatrices or other separating objects.…”

Instability of the optimal edge trajectory in the Blasius boundary layer

“…There is currently no accepted general way of choosing initial conditions for these modes, although it is expected that past some finite transient time the OTD directions naturally align with the most important directions of the system. The OTD modes have been used recently in several hydrodynamic applications, including the identification of bursting phenomena (Farazmand & Sapsis 2016), the control of linear instabilities (Blanchard, Mowlavi & Sapsis 2019) and the stability of pulsating Poiseuille flow (Kern et al 2021) as well as for faster edge tracking in high dimension (Beneitez et al 2020b). The current investigation, motivated by these promising properties, is an opportunity to test a new computational framework for stability calculations considered until now as challenging.…”