Periodic orbits near onset of chaos in plane Couette flow

Abstract

We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo an incomplete periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show that the ones that exist fall into the universal sequence described by Metropolis, Stein and Stein for unimodal maps. The periodic orbits cover much of the turbulent dynamics in that their temporal evolution overlaps with turbulent motions when projected onto a plane spanned by energy production and dissipation.

References in corpus (5)

• Lower branch coherent states in shear flows: transition and control
• Statistical analysis of coherent structures in transitional pipe flow
• Periodic motion representing isotropic turbulence
• Modeling transitional plane Couette flow
• Lifetime statistics in transitional pipe flow

Cited by in corpus (64)

• Dynamo theories
• Streamwise-localized solutions at the onset of turbulence in pipe flow
• Relative periodic orbits form the backbone of turbulent pipe flow
• Revealing the state space of turbulent pipe flow by symmetry reduction
• Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
• Spatiotemporal dynamics in 2D Kolmogorov flow over large domains
• An adjoint-based approach for finding invariant solutions of Navier-Stokes equations
• Edge states for the turbulence transition in the asymptotic suction boundary layer
• Cause-and-effect of linear mechanisms sustaining wall turbulence
• Streamwise and doubly-localised periodic orbits in plane Poiseuille flow
• Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow
• Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities
• A New Mathematical Framework for Atmospheric Blocking Events
• Crisis bifurcations in plane Poiseuille flow
• Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits
• Transition to turbulence in shear flows
• Increasing lifetimes and the growing saddles of shear flow turbulence
• Interpreted machine learning in fluid dynamics: Explaining relaminarisation events in wall-bounded shear flows
• Turbulent states in plane Couette flow with rotation
• Periodically bursting edge states in plane Poiseuille flow
• Small scale exact coherent structures at large Reynolds numbers in plane Couette flow
• Exact coherent structures and shadowing in turbulent Taylor-Couette flow
• On the probability of finding a nonphysical solution through shadowing
• Long wavelength instability of coherent structures in plane Couette flow
• Routes to turbulence in Taylor-Couette flow
• Surfing the edge: using feedback control to find nonlinear solutions
• Sustaining processes from recurrent flows in body-forced turbulence
• Studying edge geometry in transiently turbulent shear flows
• Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow
• Heteroclinic and Homoclinic Connections in a Kolmogorov-Like Flow
• Self-sustainment of coherent structures in counter-rotating Taylor-Couette flow
• Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow
• Inferring symbolic dynamics of chaotic flows from persistence
• The edge as a Lagrangian Coherent Structure in a high-dimensional state space
• Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow
• Direct path from turbulence to time-periodic solutions
• Sub critical transition to turbulence in three-dimensional Kolmogorov flow
• Linear feedback control of invariant solutions in channel flow
• Superfast amplification and superfast nonlinear saturation of perturbations as the mechanism of turbulence
• Intermittent direction reversals of moving spatially-localized turbulence observed in two-dimensional Kolmogorov flow
• Exact Coherent Structures in Fully Developed Two-Dimensional Turbulence
• Heterogeneity of the Attractor of the Lorenz '96 Model: Lyapunov Analysis, Unstable Periodic Orbits, and Shadowing Properties
• Observing a Dynamical Skeleton of Turbulence in Taylor-Couette Flow Experiments
• Streamwise decay of localized states in channel flow
• Emergence of order from chaos through a continuous phase transition in a turbulent reactive flow system
• Harbingers and latecomers - The order of appearance of exact coherent structures in plane Poiseuille flow
• Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach
• New equilibrium solution branches of plane Couette flow discovered using a project-then-search method
• Analysis and modeling of localized invariant solutions in pipe flow
• Transition in the asymptotic suction boundary layer over a heated plate
• Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow
• Resonance-Driven Mechanisms of Ion Transport and Selectivity
• A spotlike edge state in plane Poiseuille flow
• High-speed shear driven dynamos. Part 2. Numerical analysis
• Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist
• Aging and memory of transitional turbulence
• Does rare, noise-induced, bypass transition in plane Couette flow bypass instantons ?
• Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback
• Short term unpredictability of high Reynolds number turbulence --- rough dependence on initial data
• Convergence Time Towards Periodic Orbits in Discrete Dynamical Systems
• Major open problems in chaos theory and nonlinear dynamics
• The role of periodic orbits and bubbles of chaos during the transition to turbulence
• A set-oriented path following method for the approximation of parameter dependent attractors
• Dynamics-based machine learning of transitions in Couette flow