The edge as a Lagrangian Coherent Structure in a high-dimensional state space
arXiv:2004.00887 · doi:10.1103/PhysRevResearch.2.033258
Abstract
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.
arXiv comment16 pages, 10 figures, Accepted in Phys. Rev. Research
References in corpus (27)
- Dynamical multistability in high-finesse micromechanical optical cavities
- Visualizing the geometry of state space in plane Couette flow
- Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
- Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
- A Critical Comparison of Lagrangian Methods for Coherent Structure Detection
- The hidden geometry of ocean flows
- Streamwise-localized solutions at the onset of turbulence in pipe flow
- Transition in pipe flow: the saddle structure on the boundary of turbulence
- Controlling unstable chaos: Stabilizing chimera states by feedback
- Periodic orbits near onset of chaos in plane Couette flow
- Fractal Stability Border in Plane Couette Flow
- Edge States in the Climate System: Exploring Global Instabilities and Critical Transitions
- A self-sustaining nonlinear dynamo process in Keplerian shear flows
- A Theoretical Framework for Lagrangian Descriptors
- A minimization principle for the description of time-dependent modes associated with transient instabilities
- Crisis bifurcations in plane Poiseuille flow
- Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors
- Edge states as mediators of bypass transition in boundary-layer flows
- Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents
- Finding NHIM in 2 and 3 degrees-of-freedom with Hénon-Heiles type potential
- Studying edge geometry in transiently turbulent shear flows
- A critical point for bifurcation cascades and featureless turbulence
- Edge of Chaos and Genesis of Turbulence
- Edge tracking in spatially developing boundary layer flows
- Global phase space structures in a model of passive descent
- Performance of Lagrangian Descriptors and their variants in incompressible flows
- Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist