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

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.

A Trace Formula for Products of Diagonal Matrix Elements in Chaotic Systems

We derive a trace formula for $\sum_n A_{nn}B_{nn}...\delta(E-E_n)$, where $A_{nn}$ is the diagonal matrix element of the operator $A$ in the energy basis of a chaotic system. The result takes the form of a smooth term plus periodic-orbit corrections; each orbit is weighted by the usual Gutzwiller factor times $A_p B_p ...$, where $A_p$ is the average of the classical observable $A$ along the periodic orbit $p$. This structure for the orbit corrections was previously proposed by Main and Wunner (chao-dyn/9904040) on the basis of numerical evidence.Comment: 8 pages; analysis made more rigorous in the revised versio

Approach to ergodicity in quantum wave functions

According to theorems of Shnirelman and followers, in the semiclassical limit the quantum wavefunctions of classically ergodic systems tend to the microcanonical density on the energy shell. We here develop a semiclassical theory that relates the rate of approach to the decay of certain classical fluctuations. For uniformly hyperbolic systems we find that the variance of the quantum matrix elements is proportional to the variance of the integral of the associated classical operator over trajectory segments of length $T_H$, and inversely proportional to $T_H^2$, where $T_H=h\bar\rho$ is the Heisenberg time, $\bar\rho$ being the mean density of states. Since for these systems the classical variance increases linearly with $T_H$, the variance of the matrix elements decays like $1/T_H$. For non-hyperbolic systems, like Hamiltonians with a mixed phase space and the stadium billiard, our results predict a slower decay due to sticking in marginally unstable regions. Numerical computations supporting these conclusions are presented for the bakers map and the hydrogen atom in a magnetic field.Comment: 11 pages postscript and 4 figures in two files, tar-compressed and uuencoded using uufiles, to appear in Phys Rev E. For related papers, see http://www.icbm.uni-oldenburg.de/icbm/kosy/ag.htm

Semiclassical cross section correlations

We calculate within a semiclassical approximation the autocorrelation function of cross sections. The starting point is the semiclassical expression for the diagonal matrix elements of an operator. For general operators with a smooth classical limit the autocorrelation function of such matrix elements has two contributions with relative weights determined by classical dynamics. We show how the random matrix result can be obtained if the operator approaches a projector onto a single initial state. The expressions are verified in calculations for the kicked rotor.Comment: 6 pages, 2 figure

Symmetry Decomposition of Chaotic Dynamics

Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labor and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. In this paper the general formalism is developed, with the $N$-disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail.Comment: CYCLER Paper 93mar01

Semiclassical Quantization by Pade Approximant to Periodic Orbit Sums

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose a method for semiclassical quantization based upon the Pade approximant to the periodic orbit sums. The Pade approximant allows the re-summation of the typically exponentially divergent periodic orbit terms. The technique does not depend on the existence of a symbolic dynamics and can be applied to both bound and open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.Comment: 7 pages, 3 figures, submitted to Europhys. Let

On the genesis of different regimes in canopy flows: a numerical investigation

We have performed fully resolved simulations of turbulent flows over various submerged rigid canopies covering the wall of an open channel. All the numerical predictions have been obtained considering the same nominal bulk Reynolds number (i.e. Reb=UbH

A method to find unstable periodic orbits for the diamagnetic Kepler Problem

A method to determine the admissibility of symbolic sequences and to find the unstable periodic orbits corresponding to allowed symbolic sequences for the diamagnetic Kepler problem is proposed by using the ordering of stable and unstable manifolds. By investigating the unstable periodic orbits up to length 6, a one to one correspondence between the unstable periodic orbits and their corresponding symbolic sequences is shown under the system symmetry decomposition