  Dissipative dynamical systems characterised 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 automous
arbitrarily high-dimensional dissipative systems, in which the edge manifold is
re-interpreted 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 to
their scalability. Their properties are illustrated on several low-order models
of subcritical transition of increasing dimension and complexity, as well on
well-resolved 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.
