Turbulence transition and the edge of chaos in pipe flow

The linear stability of pipe flow implies that only perturbations of sufficient strength will trigger the transition to turbulence. In order to determine this threshold in perturbation amplitude we study the \emph{edge of chaos} which separates perturbations that decay towards the laminar profile and perturbations that trigger turbulence. Using the lifetime as an indicator and methods developed in (Skufca et al, Phys. Rev. Lett. {\bf 96}, 174101 (2006)) we show that superimposed on an overall $1/\Re$-scaling predicted and studied previously there are small, non-monotonic variations reflecting folds in the edge of chaos. By tracing the motion in the edge we find that it is formed by the stable manifold of a unique flow field that is dominated by a pair of downstream vortices, asymmetrically placed towards the wall. The flow field that generates the edge of chaos shows intrinsic chaotic dynamics.Comment: 4 pages, 5 figure

Correlations of electromagnetic fields in chaotic cavities

We consider the fluctuations of electromagnetic fields in chaotic microwave cavities. We calculate the transversal and longitudinal correlation function based on a random wave assumption and compare the predictions with measurements on two- and three-dimensional microwave cavities.Comment: Europhys style, 8 pages, 3 figures (included

Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the 2-fold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov-Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme we unfold a double-zero (Takens-Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (k). This scenario is extended, by the inclusion of higher order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. These waves are expected to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, thereby leaving stable upper-branch solutions whose subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.Comment: 26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mec

Echoes in classical dynamical systems

Echoes arise when external manipulations to a system induce a reversal of its time evolution that leads to a more or less perfect recovery of the initial state. We discuss the accuracy with which a cloud of trajectories returns to the initial state in classical dynamical systems that are exposed to additive noise and small differences in the equations of motion for forward and backward evolution. The cases of integrable and chaotic motion and small or large noise are studied in some detail and many different dynamical laws are identified. Experimental tests in 2-d flows that show chaotic advection are proposed.Comment: to be published in J. Phys.